Did you know? Two students can factorise 24 and 36 correctly yet choose different answers to the same word problem. One writes 12, the other 72. The difference is not multiplication skill: it is recognising whether the story is about largest equal groups or first shared repetition.
For families searching for a Secondary 1 G2 Mathematics tutor in Bukit Timah, the quick guide is: learn the divisibility tests, finish factor trees with primes, use minimum shared powers for HCF, and maximum required powers for LCM. Then return to the wording and see whether the numerical result makes sense.
The examples follow the number skills in the MOE G2 Mathematics syllabus. The eduKateSG Secondary 1 small-group tutorial reference describes a three-student weekly 90-minute teaching model near Sixth Avenue MRT, where students can practise explaining the choice of method rather than simply copying a factor tree.
HCF versus LCM in one minute
- A factor divides a number without a remainder.
- A multiple comes from multiplying a number by a positive integer.
- A prime has exactly two distinct positive factors; one is not prime.
- HCF: select the greatest factor that all given numbers share.
- LCM: select the least positive multiple shared by all numbers.
- In prime-power form, HCF uses shared minimum powers; LCM uses maximum powers across the inputs.
- For two positive integers, HCF × LCM equals their product.
The simple example that separates two similar-looking questions
Suppose a school has 24 red counters and 36 blue counters. It wants to make the greatest possible number of identical packs using every counter. The number of packs must divide both 24 and 36, so the answer is HCF(24,36) = 12. Each pack has two red and three blue counters.
Now suppose one bell rings every 24 minutes and another every 36 minutes, starting together. When do they next ring together? Their shared time must be divisible by both 24 and 36, so it is LCM(24,36) = 72 minutes. The same numbers, a different question, and a different correct answer.
A student who can explain that distinction has understood something more durable than a mnemonic. The words indicate which mathematical condition the answer must satisfy.
Sixteen ideas every Secondary 1 G2 learner should understand
1. Divisibility is a question about equal groups
A whole number a is divisible by b when dividing a by b leaves no remainder. The idea is practical: a set of 24 items can form six equal packs of four, but 25 items cannot form equal packs of four without something left over.
Example: 24 ÷ 4 = 6, with remainder zero. For 25 ÷ 4, there are six full packs and one item left over.
Learning check: Draw the packs before deciding which number is the divisor. Divisibility describes a relationship, not a button to press.
2. The test for two looks only at the final digit
A whole number is divisible by two when its final digit is even: zero, two, four, six or eight. This can be checked instantly without long division.
Example: 1,248 is even because it ends in eight; 1,247 is not divisible by two.
Learning check: Ask what happens when a number ends in zero. Zero is even, so 1,250 is divisible by two.
3. For three, the digit sum gives a quick test
A whole number is divisible by three when the sum of its digits is a multiple of three. This works even for a long number, making it a useful mental check during factorisation.
Example: For 1,326, the digit sum is 1 + 3 + 2 + 6 = 12; twelve is divisible by three, so 1,326 is divisible by three.
Learning check: The sum test does not say that the answer to 1,326 ÷ 3 is twelve. It only tells us divisibility.
4. For nine, use the same sum with a different target
Nine uses the sum of the digits, but the sum must be divisible by nine rather than merely by three. A number divisible by nine is always divisible by three, but the reverse need not hold.
Example: For 729, 7 + 2 + 9 = 18, so 729 is divisible by nine. For 123, the digit sum is six; 123 is divisible by three but not nine.
Learning check: Keep the tests distinct. ‘Multiple of three’ is insufficient for divisibility by nine.
5. The tests for five and ten are about the last digit
A whole number ending in zero or five is divisible by five. A whole number ending in zero is divisible by ten. This gives a useful way to choose which factor to remove first.
Example: 1,235 is divisible by five but not ten. 1,230 is divisible by both five and ten.
Learning check: Explain why numbers ending in five are not divisible by ten even though they contain a factor of five.
6. Four and eight use the last two or three digits
A whole number is divisible by four if its last two digits form a number divisible by four. For eight, inspect the last three digits.
Example: 3,416 is divisible by four because sixteen is divisible by four. 4,216 is divisible by eight because 216 ÷ 8 = 27.
Learning check: Do not confuse the tests: the last-two-digit rule does not determine divisibility by eight.
7. Prime numbers have exactly two positive factors
A prime number greater than one has only one and itself as positive factors. One is not prime. Two is the only even prime; every larger even number has at least three positive factors.
Example: Seven is prime because its positive factors are one and seven. Nine is composite because its factors include three.
Learning check: When listing primes, stop at numbers already proved composite. Avoid calling one prime because it has just one factor.
8. Prime factorisation decomposes without changing value
Every positive integer greater than one has a unique factorisation into primes, apart from the order of the factors. Factor trees and repeated division are two equivalent methods.
Example: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5. Multiply the prime factors back to recover sixty.
Learning check: A leaf such as six or ten is not a completed prime factor. Continue splitting until every factor is prime.
9. Exponents count repeated prime factors
An exponent states how often a factor appears in a product. A square of two is two multiplied by two, and a cube of two is three factors of two.
Example: 72 = 2³ × 3² because 2 × 2 × 2 × 3 × 3 = 72.
Learning check: Do not read 2³ as 2 × 3. Write the factors out if exponent notation is still unfamiliar.
10. HCF is the largest shared divisor
The highest common factor is the greatest positive integer that divides every quantity without remainder. It is useful for equal grouping into the largest identical units.
Example: HCF(18, 24) = 6. Eighteen and twenty-four can both be split into groups of six.
Learning check: A common factor need not be the highest. Three divides both, but six is larger.
11. LCM is the first shared repetition
The lowest common multiple is the smallest positive integer that is a multiple of each given number. It is useful for repeated events that must coincide.
Example: LCM(6, 8) = 24. Events repeating every six and eight minutes coincide again after twenty-four minutes if they start together.
Learning check: Using HCF in a repeating-cycle problem answers a different mathematical question.
12. Use prime powers systematically for HCF
When numbers are expressed as prime powers, the HCF contains only prime factors shared by all numbers, each raised to the lowest exponent among them.
Example: 24 = 2³ × 3 and 36 = 2² × 3². The shared minimum powers are 2² × 3 = 12.
Learning check: Taking the highest exponents for HCF gives an answer too large to divide both originals.
13. Use the largest required prime powers for LCM
The LCM must contain enough of every prime factor to be divisible by both numbers. Choose the highest exponent of each prime appearing in either factorisation.
Example: With 24 = 2³ × 3 and 36 = 2² × 3², LCM = 2³ × 3² = 72.
Learning check: Using only the shared primes misses factors required by one of the numbers.
14. A useful two-number check links HCF and LCM
For positive integers a and b, HCF(a,b) multiplied by LCM(a,b) equals a × b. This identity is a useful numerical check, but it does not replace understanding the original question.
Example: For 18 and 24, HCF is six and LCM is seventy-two. Their product is 432, matching 18 × 24.
Learning check: Use this check for two positive integers; do not generalise the product identity unchanged to three or more numbers.
15. Perfect squares and cubes can be recognised by prime exponents
In a perfect square’s prime factorisation, all exponents are even. In a perfect cube, every exponent is a multiple of three. This links factorisation to roots.
Example: 144 = 2⁴ × 3² is a perfect square, with positive square root 12. 216 = 2³ × 3³ is a perfect cube, with cube root six.
Learning check: A positive integer with one unpaired prime factor is not a perfect square.
16. Word problems reveal which operation is meaningful
The arithmetic may be straightforward; identifying the model is the demanding step. ‘Largest identical group with no leftovers’ suggests HCF. ‘Next time together’ suggests LCM.
Example: Two alarms ring every twelve and eighteen minutes. Starting together, they next ring together after LCM(12,18) = 36 minutes.
Learning check: The word ‘largest’ alone does not always mean HCF. Explain the quantities and the event being modelled.
26 quick worked clinics
Try each question before reading its answer. A good solution should show why the number fits the definition or context, not just a calculator output.
Clinic 1: Is 1,458 divisible by 2?
Answer: Yes. Why: Last digit eight is even.
Verify: Check the final digit. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 2: Is 1,458 divisible by 3?
Answer: Yes. Why: Digit sum 1+4+5+8 = 18.
Verify: Eighteen is a multiple of three. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 3: Is 1,458 divisible by 9?
Answer: Yes. Why: Digit sum is eighteen, divisible by nine.
Verify: This is a stronger test than divisibility by three. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 4: Is 1,245 divisible by 5?
Answer: Yes. Why: Final digit is five.
Verify: It is not divisible by ten. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 5: Is 1,240 divisible by 10?
Answer: Yes. Why: Final digit zero.
Verify: A final zero also satisfies divisibility by five and two. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 6: Is 3,416 divisible by 4?
Answer: Yes. Why: Last two digits sixteen divide by four.
Verify: Do not inspect only the final digit. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 7: Is 4,216 divisible by 8?
Answer: Yes. Why: Last three digits 216 are divisible by eight.
Verify: 216 ÷ 8 = 27. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 8: Is 123 divisible by 9?
Answer: No. Why: Digits total six.
Verify: 123 is divisible by three, not nine. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 9: Is 1 a prime number?
Answer: No. Why: One has only one positive factor.
Verify: A prime has exactly two positive factors. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 10: What is the smallest prime number?
Answer: 2. Why: Two has positive factors one and two.
Verify: It is the only even prime. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 11: Factorise 36 into primes
Answer: 2² × 3². Why: 36 = 4 × 9.
Verify: Multiply factors back. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 12: Factorise 84 into primes
Answer: 2² × 3 × 7. Why: 84 = 4 × 21.
Verify: Every listed factor must be prime. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 13: Factorise 120 into primes
Answer: 2³ × 3 × 5. Why: 120 = 8 × 15.
Verify: Check product returns 120. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 14: Factorise 180 into primes
Answer: 2² × 3² × 5. Why: 180 = 4 × 9 × 5.
Verify: Check all leaves prime. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 15: Find HCF(18,24)
Answer: 6. Why: Six is greatest common divisor.
Verify: Both divided by six leave no remainder. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 16: Find LCM(18,24)
Answer: 72. Why: Smallest number divisible by 18 and 24.
Verify: Check 72/18 = 4 and 72/24 = 3. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 17: Find HCF(24,36)
Answer: 12. Why: Shared minimum prime powers 2² × 3.
Verify: Do not use maximum powers. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 18: Find LCM(24,36)
Answer: 72. Why: Maximum prime powers 2³ × 3².
Verify: Check both divide 72. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 19: Find HCF(60,72)
Answer: 12. Why: 60 = 2² × 3 × 5; 72 = 2³ × 3².
Verify: Shared minimum factors are 2² × 3. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 20: Find LCM(60,72)
Answer: 360. Why: Use 2³ × 3² × 5.
Verify: Check 360/60 = 6; 360/72 = 5. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 21: Largest identical packs from 24 red and 36 blue items?
Answer: 12 packs. Why: HCF of 24 and 36 is twelve.
Verify: Each pack has two red and three blue. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 22: Alarms ring every 8 and 12 minutes. Next simultaneous ring?
Answer: 24 minutes. Why: LCM(8,12) = 24.
Verify: Twenty-four is a multiple of both periods. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 23: Two ribbons 45 cm and 60 cm. Longest equal cut with no waste?
Answer: 15 cm. Why: HCF(45,60) = 15.
Verify: Each ribbon is split into whole segments. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 24: Find positive square root of 144
Answer: 12. Why: Twelve squared is 144.
Verify: Factorisation is 2⁴ × 3². If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 25: Find cube root of 216
Answer: 6. Why: Six cubed is 216.
Verify: Prime factorisation is 2³ × 3³. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
Clinic 26: Is 72 a perfect square?
Answer: No. Why: 72 = 2³ × 3²; exponent three is odd.
Verify: Not every prime exponent is even. If this question exposed a misunderstanding, revise that idea and then test a changed version a few days later.
A practical four-week learning plan
Week 1: see factors and multiples as different sets
List divisors and multiples of small numbers, use everyday grouping examples, and practise divisibility by two, three, four, five, eight, nine and ten. Compare which tests answer yes or no for the same number.
Use a small error ledger to record whether the mistake came from arithmetic, incomplete prime factorisation or selecting the wrong relationship. Short, corrected practice can be more valuable than another large worksheet done without feedback.
Week 2: make prime factorisation dependable
Use factor trees and repeated prime division. Finish only when every factor is prime, then multiply back to check. Introduce index notation once the repeated-factor meaning is secure.
Use a small error ledger to record whether the mistake came from arithmetic, incomplete prime factorisation or selecting the wrong relationship. Short, corrected practice can be more valuable than another large worksheet done without feedback.
Week 3: derive HCF and LCM from prime factors
Practise pairs such as 18 and 24, then 60 and 72. Ask why HCF uses minimum shared powers while LCM uses maximum needed powers. Check with division and the two-number product identity.
Use a small error ledger to record whether the mistake came from arithmetic, incomplete prime factorisation or selecting the wrong relationship. Short, corrected practice can be more valuable than another large worksheet done without feedback.
Week 4: distinguish the stories
Mix contexts about equal packs, common cycles, cutting lengths and repeated events. Have the child write the required condition before choosing an operation. Finish with unfamiliar mixed problems without chapter labels.
Use a small error ledger to record whether the mistake came from arithmetic, incomplete prime factorisation or selecting the wrong relationship. Short, corrected practice can be more valuable than another large worksheet done without feedback.
What parents can check on a marked worksheet
When a HCF or LCM question is wrong, do not automatically assign more division sums. First ask which number must divide which quantity. If the child gives LCM when the story asks for identical packs, the error is in modelling. If the correct method is chosen but the factor tree contains a composite leaf, the repair is in factorisation.
Ask the learner to reconstruct the method on fresh numbers and confirm with an independent check. For a proposed HCF, can it divide both originals? For a proposed LCM, are both originals factors of it? If the child can justify these without prompts, understanding is becoming more stable.
Frequently asked questions
What is the difference between HCF and LCM?
HCF is the largest number dividing all given quantities. LCM is the smallest positive number divisible by all the given quantities. Equal-grouping contexts often call for HCF; synchronised repeats usually call for LCM.
How do I teach divisibility by three?
Add the digits and check whether the sum is divisible by three. Explain that this is a divisibility test, not the final quotient.
Why is one not prime?
By definition a prime number has exactly two distinct positive factors. One has only one positive factor.
Can a calculator do all the factorisation work?
It can check products and quotients, but the student still needs to represent prime factors and choose HCF or LCM appropriately for the question.
Does G2 Maths use the same topic order as G3?
The subject levels share fundamentals but their syllabus progression and requirements differ. Follow the student’s current G2 scheme of work.
Why does my child reverse highest and lowest powers?
They may be memorising a rule without seeing why a factor must divide both original numbers or contain enough prime factors to be a multiple of both. Use small numerical examples first.
How do we check an LCM answer?
Divide the proposed LCM by each given number; the results should be whole numbers. Then consider whether any smaller positive common multiple exists.
How much practice is useful?
Start with short mixed sets comparing factors, multiples and word-problem interpretations. Review recurring errors and retest independently after a gap.
Can a three-student tutorial help with these misconceptions?
A small-group tutor can compare different factor trees, inspect HCF/LCM choices and provide independent transfer questions. The source eduKateSG programme describes weekly 90-minute three-pupil sessions near Sixth Avenue MRT.
Trusted references and related mathematics guides
Consult the MOE G2 Mathematics syllabus for curriculum context and the eduKateSG Secondary 1 small-group tutoring reference for the teaching setting.
- Negative numbers, fractions and estimation
- Algebra and equations after number fundamentals
- Choosing a method in word problems
Did you know that checking whether an answer satisfies a condition is itself a form of mathematical reasoning? For HCF and LCM, that question can separate a memorised procedure from genuine understanding.