Secondary 2 G3 Mathematics Tutor Bukit Timah | Algebraic Proofs with Consecutive Integers and Divisibility. Did you know? Checking that 2+3=5 and 7+8=15 shows two examples of a pattern, but does not prove it works for every pair of consecutive whole numbers. Algebraic proof explains the whole family of cases at once.
This Bukit Timah Secondary 2 G3 Mathematics guide is about turning a numerical observation into a general statement and a sound argument. We begin with the meanings of even, odd, consecutive, divisible and counterexample; then practise original small proofs. Some proofs are straightforward algebraic consolidations and others are clearly described as extension for pupils ready for formal reasoning beyond the school’s immediate Secondary 2 test.
A successful proof does not need fancy language. It needs a correct general representation, valid algebra, and a final statement that answers the claim. A small-group tutor can ask every student to explain why the same calculation covers all permitted integers instead of just one lucky example.
Find the proof question you need
- Quick answer: represent the integer generally
- Testing examples versus proving a universal claim
- Even and odd are useful algebraic forms
- A proof of divisibility must show an integer factor
- Consecutive integers create useful parity patterns
- Proof and converse are separate claims
- Choose the right variable from the claim
- How to write the final line of an algebraic proof
- Forty original algebraic proof and counterexample clinics
- Even plus even is even
- Even times any integer is even
- Consecutive product even
- Even prime exception
- Proof with a square of a sum
- Even square implies even root
- A proof that one counterexample is enough
- A 20-minute proof clinic for three students
- A two-week progression into general mathematical language
- Frequently asked: does substituting many values count as a proof?
- Frequently asked: is this too advanced for Secondary 2?
- Frequently asked: why should we use two different letters for two odd numbers?
- Frequently asked: what makes a counterexample valid?
- Frequently asked: should a proof end with a sentence?
- Published Bukit Timah programme and connected learning
Quick answer: represent the integer generally
Let n be any integer. An even integer can be written 2n; an odd integer can be written 2n+1. Consecutive integers are n, n+1, n+2 and so on. To prove a quantity is divisible by k, rewrite it as k × (an integer). To disprove a universal claim, one valid counterexample is enough.
For example, the sum of three consecutive integers is n+(n+1)+(n+2)=3n+3=3(n+1). Since n+1 is an integer, the sum is divisible by 3 for every integer n. This proof covers positive, zero and negative starts without listing separate trials.
Testing examples versus proving a universal claim
Examples can make a conjecture plausible and can quickly find a counterexample to a false claim. But ten true examples do not establish a statement for every integer. A proof works with a general integer and uses an identity or structural property that holds regardless of its value.
Suppose someone says ‘the square of every integer is greater than the original integer’. Checking 2 and 3 supports it, but n=0 gives 0²=0, not greater, and n=−1 gives 1>−1. The single example n=0 disproves the universal statement as written. Defining the domain and paying attention to strict words like ‘greater’ matters.
Even and odd are useful algebraic forms
An even number is a multiple of 2, so write 2k where k is an integer. An odd number is one more than an even number, so write 2k+1. Two different odd integers should normally use different letters, such as 2a+1 and 2b+1, unless their relation is specified.
For the sum of two odd integers, (2a+1)+(2b+1)=2(a+b+1), which is twice an integer and therefore even. The argument is more general than checking 3+5 and 7+9, and it avoids assuming that both odd values must be consecutive.
A proof of divisibility must show an integer factor
To show an expression is divisible by 6, it is enough to rewrite it as 6 multiplied by an integer. For n(n+1)(n+2), the product of three consecutive integers, at least one factor is divisible by 3 and at least one is even, so the product has factors 2 and 3 and is divisible by 6.
The factor argument uses the fact that 2 and 3 are coprime. It is not enough to show that a number is divisible by 4 and by 6 and then automatically claim divisibility by 24; their least common multiple is 12, not 24. Precise factor structure matters.
Consecutive integers create useful parity patterns
Two consecutive integers n and n+1 always have opposite parity: if n is even, n+1 is odd; if n is odd, n+1 is even. Hence their product is even. A stronger pattern is that the product of any three consecutive integers is divisible by 6, and the product of four consecutive integers is divisible by 24.
The last statement is a useful extension: among four consecutive integers, one is divisible by 4, another even factor supplies at least one more 2, and one is divisible by 3. Thus 2³×3=24 divides their product. Ask a pupil to explain the factor availability rather than copy the divisibility conclusion.
Proof and converse are separate claims
If a figure is a square, it is a rectangle. The converse ‘if a figure is a rectangle, it is a square’ is false: a 3 cm by 5 cm rectangle is a counterexample. Knowing one direction of implication does not guarantee the reverse. This distinction also appears in algebra and number properties.
A good proof question specifies which direction it needs. Showing an even number has an even square establishes one implication. To prove the converse ‘if a square is even, the original integer is even’, you need a separate argument—such as observing that an odd integer’s square is odd.
Choose the right variable from the claim
- If the claim concerns every integer, write n∈ℤ, not just a positive example.
- If the claim concerns two arbitrary odd integers, write 2a+1 and 2b+1 to allow different values.
- If the claim concerns consecutive integers, write n and n+1; for three, add n+2.
- If the claim concerns consecutive even numbers, write 2n and 2n+2, not n and n+2 unless n was declared even.
- If the claim concerns multiples of k, write kn with integer n.
- After simplifying, point explicitly to the factor that proves divisibility or to the counterexample that disproves the claim.
How to write the final line of an algebraic proof
The final line should connect the algebraic form with the stated conclusion. After writing n(n+1)=2k for some integer k, say ‘therefore n(n+1) is even’. After obtaining 3(n+1), say ‘since n+1 is an integer, this expression is divisible by 3’. If the claim is false, state which permitted value violates it and why.
Do not simply end at 2(n+1) and expect the reader to guess the meaning. A proof is a short explanation, not only a rearranged expression.
Forty original algebraic proof and counterexample clinics
Odd plus odd is even
Statement to examine. Prove the sum of any two odd integers is even.
General proof or counterexample. Write 2a+1 and2b+1. Sum=2a+2b+2=2(a+b+1).
Why this settles the claim. The result is twice an integer. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Checking only3+5 does not prove the universal case.
Independent check or variation. Try a=−2,b=5 to show proof also covers negative odd numbers. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Even plus even is even
Statement to examine. Prove the sum of two arbitrary even integers is even.
General proof or counterexample. Use 2a+2b=2(a+b).
Why this settles the claim. The shared factor two establishes evenness. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Do not assume the numbers are equal.
Independent check or variation. Any choices of integers a and b work. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Odd plus even is odd
Statement to examine. Prove one odd plus one even integer is odd.
General proof or counterexample. (2a+1)+2b=2(a+b)+1.
Why this settles the claim. It is one more than a multiple of two. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Writing 2(a+b+1) would incorrectly make it even.
Independent check or variation. At a=1,b=2 the sum3+4=7. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Odd times odd is odd
Statement to examine. Prove product of two odd integers is odd.
General proof or counterexample. (2a+1)(2b+1)=4ab+2a+2b+1=2(2ab+a+b)+1.
Why this settles the claim. The expanded product is twice an integer plus one. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Do not claim multiplying numbers always preserves their signs or parity without structure.
Independent check or variation. Test −3×5=−15, still odd. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Even times any integer is even
Statement to examine. Let e=2a and k any integer. Prove ek even.
General proof or counterexample. ek=(2a)k=2(ak).
Why this settles the claim. The product retains factor two regardless of k. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Requiring k itself to be even is unnecessary.
Independent check or variation. At k=−7 the product remains even. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Square of an even integer is divisible by four
Statement to examine. Let n be even. Prove n² multiple4.
General proof or counterexample. n=2k gives n²=4k².
Why this settles the claim. k² remains an integer. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Saying only ‘even square is even’ is weaker than required.
Independent check or variation. For n=6, n²=36=4×9. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Square of an odd integer is odd
Statement to examine. Let n=2k+1. Show n² odd.
General proof or counterexample. n²=4k²+4k+1=2(2k²+2k)+1.
Why this settles the claim. The expansion has form2m+1. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Dropping the cross term 4k produces an invalid expansion.
Independent check or variation. For n=−3, square9 is odd. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Consecutive product even
Statement to examine. Prove n(n+1) is even for every integer n.
General proof or counterexample. Two consecutive integers include an even one, so their product is even.
Why this settles the claim. Parity alternates between consecutive integers. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Both factors need not be even.
Independent check or variation. n=3 gives12; n=4 gives20. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Algebraic form for consecutive product
Statement to examine. Show n²+n is always even.
General proof or counterexample. n²+n=n(n+1), a product of consecutive integers.
Why this settles the claim. At least one factor divisible by2. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. A few even outputs are observations, not proof.
Independent check or variation. This holds for n negative or zero too. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Sum of three consecutive divisible by three
Statement to examine. Prove n+(n+1)+(n+2) divisible3.
General proof or counterexample. Sum=3n+3=3(n+1).
Why this settles the claim. n+1 is an integer. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. If using only1+2+3, the general case is not established.
Independent check or variation. At n=−2 the sum is−3. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Sum of five consecutive divisible by five
Statement to examine. Prove sum n through n+4 divisible5.
General proof or counterexample. Sum=5n+10=5(n+2).
Why this settles the claim. Factor five from the whole expression. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Focusing only on positive n narrows the claim unnecessarily.
Independent check or variation. For n=1 sum15, and15 divisible5. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Three odd consecutive numbers
Statement to examine. Show sum of three consecutive odd integers is divisible by3.
General proof or counterexample. Represent 2n+1,2n+3,2n+5. Sum=6n+9=3(2n+3).
Why this settles the claim. The three odds differ by two. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Using n,n+1,n+2 does not ensure all odd.
Independent check or variation. At n=0 the numbers1,3,5 sum9. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Product of three consecutive integers divisible six
Statement to examine. Prove n(n+1)(n+2) divisible6.
General proof or counterexample. Among three consecutive, one multiple of3 and at least one even, so product contains 2×3=6.
Why this settles the claim. The factors2 and3 come from the same product. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Showing only an even factor proves divisibility2, not6.
Independent check or variation. At n=4, product4×5×6=120. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Difference between consecutive squares is odd
Statement to examine. Find (n+1)²−n² and show odd.
General proof or counterexample. Expand n²+2n+1−n²=2n+1.
Why this settles the claim. The result has odd form for integer n. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Difference of squares is not always even.
Independent check or variation. For n=5 the difference36−25=11. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Sum of consecutive squares is odd
Statement to examine. Prove n²+(n+1)² odd.
General proof or counterexample. Expand 2n²+2n+1=2n(n+1)+1.
Why this settles the claim. Twice an integer plus one is odd. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Do not confuse with the difference of squares statement.
Independent check or variation. At n=2, 4+9=13. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Odd square minus one divisible eight
Statement to examine. Prove (2n+1)²−1 divisible8.
General proof or counterexample. Expand4n²+4n=4n(n+1); consecutive product is even, so expression multiple8.
Why this settles the claim. The extra factor two comes from n(n+1). The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Only writing ‘multiple4’ misses the stronger claim.
Independent check or variation. For odd7,49−1=48=8×6. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Squares of even numbers not always divisible eight
Statement to examine. Disprove ‘every even square is a multiple of8’.
General proof or counterexample. Take n=2: n²=4, which is not divisible by8.
Why this settles the claim. One valid counterexample refutes a universal claim. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Checking only n=4 would misleadingly support it.
Independent check or variation. n=6 has square36, also not divisible by8. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Even prime exception
Statement to examine. Disprove ‘every prime number is odd’.
General proof or counterexample. The number2 is prime and even.
Why this settles the claim. Two has exactly two positive divisors, 1 and2. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. One is not prime, so it is not a relevant exception.
Independent check or variation. Every even integer greater than2 has factor2 and is composite. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
A rectangle is not always a square
Statement to examine. Disprove converse ‘every rectangle is a square’.
General proof or counterexample. A 3-by5 cm rectangle has four right angles but unequal adjacent sides.
Why this settles the claim. It meets rectangle conditions but fails square’s equal-side condition. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. The original implication square→rectangle is still true.
Independent check or variation. A 4-by4 rectangle is a square, but does not prove converse. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
A square is a rectangle
Statement to examine. Prove any square is a rectangle.
General proof or counterexample. By definition a square has four right angles and four equal sides, so it satisfies rectangle’s right-angle property.
Why this settles the claim. The stronger set of square properties includes rectangle conditions. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. The converse does not automatically follow.
Independent check or variation. Draw a non-square rectangle to distinguish. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Two consecutive even numbers product divisible eight
Statement to examine. Prove (2n)(2n+2) multiple8.
General proof or counterexample. Product4n(n+1); since n(n+1) even, product divisible8.
Why this settles the claim. The two adjacent even numbers supply more than just factor4. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Saying only multiple4 is correct but weaker.
Independent check or variation. For n=1, 2×4=8; for n=2,4×6=24. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
A multiple of three has square multiple nine
Statement to examine. Let n=3k. Prove n² divisible9.
General proof or counterexample. n²=9k².
Why this settles the claim. The coefficient nine is explicit. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Being a multiple3 alone does not make n itself multiple9.
Independent check or variation. At n=6 square36=9×4. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
A number divisible by six is divisible by three
Statement to examine. Prove using n=6k.
General proof or counterexample. n=6k=3(2k).
Why this settles the claim. 2k remains integer. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. The converse is false: three can be divisible by3 but not6.
Independent check or variation. Also n=2(3k), so it is even. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
A multiple of four is even
Statement to examine. Let n=4k. Prove n multiple2.
General proof or counterexample. n=2(2k).
Why this settles the claim. Every multiple of4 contains factor2. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. The reverse ‘every even integer multiple4’ is false; choose n=2.
Independent check or variation. At n=12 both properties hold. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Difference of odd integers is even
Statement to examine. Let a=2m+1,b=2n+1. Show a−b even.
General proof or counterexample. a−b=2(m−n).
Why this settles the claim. Subtracting the +1 terms leaves a factor2. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. The difference can be negative but remains even.
Independent check or variation. At a=3,b=9 difference−6. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Difference of consecutive odd squares multiple eight
Statement to examine. Numbers 2n+1 and2n+3 are consecutive odd integers.
General proof or counterexample. Difference (2n+3)²−(2n+1)² =8n+8=8(n+1).
Why this settles the claim. Difference-of-squares or expansion exposes factor8. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. The claim concerns consecutive odd values, not arbitrary odd squares.
Independent check or variation. At 3 and5,25−9=16. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
A common factor in algebra
Statement to examine. Show 3n+6 is divisible3 for every integer n.
General proof or counterexample. 3n+6=3(n+2).
Why this settles the claim. The term in brackets is an integer. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. This is not necessarily divisible6 for odd n.
Independent check or variation. At n=1, value9. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
n squared minus n is even
Statement to examine. Prove n²−n divisible2.
General proof or counterexample. Factor n(n−1), a product of consecutive integers.
Why this settles the claim. One factor is even. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. A test n=5 gives20 but does not prove all.
Independent check or variation. Works at n=0 and negative integers. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
n cubed minus n multiple six
Statement to examine. Prove n³−n divisible6.
General proof or counterexample. Factor n(n−1)(n+1), product of three consecutive integers.
Why this settles the claim. It contains an even factor and multiple of3. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Claim does not require n positive.
Independent check or variation. At n=4,64−4=60. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
A square divisible by three?
Statement to examine. Disprove ‘every perfect square is divisible by3’.
General proof or counterexample. Take 2²=4, not divisible by3.
Why this settles the claim. One counterexample resolves the false universal statement. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. 1²=1 is another counterexample.
Independent check or variation. A square of a multiple3, such as6², is divisible9. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Sum of two consecutive integers always odd
Statement to examine. Prove n+(n+1) odd.
General proof or counterexample. Sum=2n+1.
Why this settles the claim. Odd form is immediate. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. The product of the pair is even, a different property.
Independent check or variation. At n=−1, −1+0=−1, odd. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Sum of four consecutive integers even
Statement to examine. Prove n+(n+1)+(n+2)+(n+3) is even.
General proof or counterexample. Sum=4n+6=2(2n+3).
Why this settles the claim. A factor of two is explicit. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Four consecutive sum is not always divisible4, e.g.1+2+3+4=10.
Independent check or variation. At n=2 sum14. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Two consecutive numbers cannot both be even
Statement to examine. Show why n,n+1 are not both even.
General proof or counterexample. If n=2k, then n+1=2k+1 odd; if n odd, n+1 even.
Why this settles the claim. Parity alternates between consecutive integers. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. A single example alone does not provide the general proof.
Independent check or variation. Exactly one of the pair is even. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Product of two consecutive integers divisible by two
Statement to examine. Represent any consecutive pair and prove result.
General proof or counterexample. n(n+1)=2m for some integer m, because one factor is even.
Why this settles the claim. The proof uses divisibility of a product, not addition. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Don’t claim both factors divisible2.
Independent check or variation. Example9×10=90 supports but does not prove. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
A general odd expression
Statement to examine. Show 2n+7 odd for integer n.
General proof or counterexample. 2n+7=2(n+3)+1.
Why this settles the claim. The expression is one more than an even multiple. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Not every positive odd expression is a prime.
Independent check or variation. At n=−2, expression3. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
An expression always even
Statement to examine. Prove 6n+4 even.
General proof or counterexample. 6n+4=2(3n+2).
Why this settles the claim. The bracket is integer. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Even does not mean divisible4 in every case.
Independent check or variation. At n=1, value10. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
An expression divisible by five
Statement to examine. Show 10n+15 multiple5.
General proof or counterexample. 10n+15=5(2n+3).
Why this settles the claim. Factor5 explicitly. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. It need not be multiple10.
Independent check or variation. At n=0 value15. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
A counterexample to sum of squares even
Statement to examine. Disprove ‘the sum of squares of any two integers is even’.
General proof or counterexample. Take 1²+2²=1+4=5, odd.
Why this settles the claim. A single valid example makes the universal claim false. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. If both original integers share parity, the sum is even, but not always.
Independent check or variation. Use 2 and3 to get4+9=13. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Counterexample to every square positive
Statement to examine. Disprove ‘n² is always strictly positive for integer n’.
General proof or counterexample. Choose n=0: n²=0, not greater than0.
Why this settles the claim. Nonnegative is the correct universal statement. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Confusing positive with nonnegative changes the claim.
Independent check or variation. All nonzero integer squares are positive. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
A true converse about even squares
Statement to examine. Show if n² is even then n is even.
General proof or counterexample. If n were odd, n=2k+1 gives n²=2(2k²+2k)+1 odd, contradiction.
Why this settles the claim. An odd integer cannot have an even square. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Assuming the converse without a separate reason is not a proof.
Independent check or variation. Together with even→even square, this gives an equivalence. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Three multiples of four
Statement to examine. Prove sum of any three multiples4 divisible4.
General proof or counterexample. Write 4a+4b+4c=4(a+b+c).
Why this settles the claim. The shared factor4 remains under addition. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. The three integers a,b,c need not be consecutive.
Independent check or variation. At 4,8,12 sum24. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Odd squared equals one modulo eight
Statement to examine. Prove odd squares leave remainder1 when divided by8.
General proof or counterexample. Let odd n=2k+1. n²−1=4k(k+1), and k(k+1) even, so multiple8.
Why this settles the claim. The preceding factor argument applies for every integer k. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Checking only 3²=9 and5²=25 is insufficient.
Independent check or variation. 7²=49, remainder1. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
A false claim about multiples
Statement to examine. Disprove ‘if n is divisible by2 then n is divisible by4’.
General proof or counterexample. n=2 is divisible2 but not4.
Why this settles the claim. One counterexample separates necessary and sufficient conditions. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. If divisible4 then divisible2 is true; converse is not.
Independent check or variation. n=6 is another counterexample. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Proof with a square of a sum
Statement to examine. Show (n+1)²−n² odd.
General proof or counterexample. (n²+2n+1)−n²=2n+1.
Why this settles the claim. The difference of consecutive squares is twice an integer plus one. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. The adjacent integers need not be positive.
Independent check or variation. At n=−2, difference(−1)²−(−2)²=−3. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Six divides the product of three consecutive
Statement to examine. Show (n−1)n(n+1) multiple6.
General proof or counterexample. Three consecutive integers include a multiple3 and an even integer.
Why this settles the claim. Their product contains factors2 and3. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. The middle integer n need not itself be even or divisible3.
Independent check or variation. At n=5,4×5×6=120. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Expression not divisible by two universally
Statement to examine. Disprove claim n²+2n always even.
General proof or counterexample. Choose n=1: 1+2=3, odd.
Why this settles the claim. The expression is n(n+2), same parity as n. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. A few even-n checks conceal odd counterexamples.
Independent check or variation. At n=2, value8 is even but not universal. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
A fact about four consecutive numbers
Statement to examine. Show their product divisible24 (extension).
General proof or counterexample. Among four consecutive are a multiple4, another even number and a multiple3; combined factors include 8×3=24.
Why this settles the claim. Factor divisibility establishes the claim without multiplying full polynomial. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Just checking1×2×3×4=24 is not a universal proof.
Independent check or variation. At 2,3,4,5 product120 divisible24. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
A non-integer domain warning
Statement to examine. Claim n(n+1) is even. Does it apply to all real n?
General proof or counterexample. No; ‘even’ concerns integers. For n=1/2, product3/4 is not an even integer.
Why this settles the claim. The domain is an essential part of the theorem. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Extending a theorem for integers to all reals changes its meaning.
Independent check or variation. Always write n∈ℤ in the proof. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
A numerical pattern needing proof
Statement to examine. Values of n²+n at n=1,2,3 are2,6,12. What proves evenness?
General proof or counterexample. Factor n(n+1); one consecutive factor is even.
Why this settles the claim. The algebraic form covers all integer n. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. The three listed values are supporting examples, not proof.
Independent check or variation. Try n=−4; product12 still even. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
A statement with wrong direction
Statement to examine. If a number is a multiple12, it is a multiple6. Is the converse true?
General proof or counterexample. Forward true because12k=6(2k). Converse false:6 is multiple6 but not12.
Why this settles the claim. A conditional statement and its converse must be tested separately. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. One true direction cannot certify the other.
Independent check or variation. The counterexample n=18 also works. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
Even square implies even root
Statement to examine. Given integer n and n² divisible by2, show n even.
General proof or counterexample. An odd n would make odd square, so n cannot be odd; integer n must be even.
Why this settles the claim. The proof uses parity alternatives. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. Do not take real square roots and ignore integrality.
Independent check or variation. n²=36 yields n=±6, both even. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
A proof that one counterexample is enough
Statement to examine. Claim all prime numbers are greater than two. Evaluate.
General proof or counterexample. False: 2 itself is prime and not greater than two.
Why this settles the claim. The universal quantifier is disproved by one permitted exception. The explanation must cover every integer allowed by the statement, or give one genuine exception when the statement claims universality.
Common logical mistake. The statement all primes at least two is true.
Independent check or variation. List 2,3,5 to compare wording. Ask the learner to write the domain and give a complete concluding sentence, not only an algebraic expression.
A 20-minute proof clinic for three students
Write three claims on the board: ‘The sum of two odd integers is even’, ‘the product of two consecutive integers is even’, and ‘every even integer is divisible by four’. One pupil builds the general algebraic proof of the first, another uses parity on consecutive integers for the second, and the third searches for a counterexample to the false third. All pupils then rotate through the tasks.
The comparison illustrates the difference between proving and disproving. The two true statements require an argument covering infinitely many allowed integers. The false statement is settled by a single permitted exception, such as 2. This is a more useful lesson than memorising a list of numerical patterns without explaining why any must hold.
A two-week progression into general mathematical language
Week one: Practise forming arbitrary even and odd numbers, consecutive integers and multiples of k. Expand two or three short expressions and factor the result into 2×integer or 3×integer. Insist on precise conclusions. Finish with one false universal statement for which the pupil must find and explain a counterexample.
Week two: Introduce differences of consecutive squares, product divisibility and the distinction between a statement and its converse. Use no-heading mixed questions: is the task to prove, disprove, simplify or solve? The last activity is to write a short proof independently, then correct a friend’s valid-looking but incomplete argument.
Frequently asked: does substituting many values count as a proof?
Usually not for a statement about every integer. Testing many cases can suggest a pattern, and it can disprove a false universal claim when a counterexample is found. To establish the statement for all integers, use general representations and valid logic. An algebraic expression with a common factor often provides a concise proof.
Frequently asked: is this too advanced for Secondary 2?
The basic algebraic forms of even, odd and consecutive numbers build directly on lower-secondary number and algebra skills. Some stronger divisibility results may be enrichment or later-year content. A tutor should use the school’s taught topics and the pupil’s fluency to select suitable depth. The aim is clear reasoning, not early coverage for its own sake.
Frequently asked: why should we use two different letters for two odd numbers?
Because two arbitrary odd integers may be different values. Writing 2n+1 twice would represent the same number two times, proving only a special case. Writing 2a+1 and2b+1 allows any pair of odd integers, which is what the claim requires.
Frequently asked: what makes a counterexample valid?
It must satisfy every condition in the statement and violate the conclusion. To disprove ‘every even integer divisible by four’, choose 2: it is even and not divisible by four. A counterexample outside the domain, such as one-half when the claim concerns integers, would not disprove the claim.
Frequently asked: should a proof end with a sentence?
Yes. After factoring n(n+1) or 2(a+b+1), connect the result to the requested property. A short line such as ‘Since a+b+1 is an integer, the sum is even’ identifies the logical bridge. It makes the solution verifiable and teaches the student to distinguish calculation from argument.
Published Bukit Timah programme and connected learning
The unchanged eduKateSG small-group Mathematics benchmark describes three-pupil tutorials of around 1.5 hours weekly, first-principles teaching and guided corrections at 8 Fourth Avenue, Singapore 268674 near Sixth Avenue MRT. The Bukit Timah title identifies nearby families, not additional teaching premises or class availability.
G3 is a Mathematics subject level under Full Subject-Based Banding. The MOE G2/G3 teaching syllabus provides progression context. Some number-theory statements in this article are enrichment; the immediate Secondary 2 school assessment scope determines priority.
Related guides: number patterns and sequences, algebraic identities, and showing and checking mathematical working. These develop complementary skills rather than duplicate this page’s proof and counterexample focus.
For a Secondary 2 G3 Maths consultation near Bukit Timah, review the published programme reference and bring one proof or number-pattern question that the pupil could not explain. Ask about group fit, current class times and places before visiting.
Editorial review: 27 October 2026. All cases are original teaching illustrations. Unless stated otherwise, n, a, b and k denote integers; algebraic counterexamples must remain within the stated domain.