Secondary 1 G3 Mathematics Tutor Bukit Timah | Prime Factorisation, HCF and LCM

Did you know? The same two numbers can tell two completely different stories. When 48 pencils and 60 erasers must be divided into the greatest possible number of identical gift packs, you look for a highest common factor. When two lights flash every 12 and 18 seconds and you want to know when they flash together again, you look for a lowest common multiple. Both use factors and multiples, but the questions ask for different things.

This Secondary 1 G3 Mathematics tutor guide for Bukit Timah families makes prime factorisation, HCF and LCM understandable from first principles. We begin with the meanings, not memorised procedures, then apply the ideas to worked examples, square roots, cube roots and typical mistakes in school questions.

Quick answer: which method do you need?

HCF is the largest positive whole number that divides each given number exactly. LCM is the smallest positive whole number that is a multiple of every given number. Prime factorisation helps you calculate both reliably.

If the question asks for the greatest number of equal groups with no leftovers, HCF is often relevant. If several repeating events must coincide, LCM is often relevant. These clues are useful, but read the actual conditions: a question containing the word “groups” is not automatically an HCF question.

Take 12 and 18. Their HCF is 6, because 6 is the largest number that divides both. Their LCM is 36, because 36 is the first positive common multiple. We will show why rather than ask you to accept those numbers.

What makes a number prime?

A prime number is a whole number greater than 1 with exactly two distinct positive factors: 1 and the number itself. For example, 2, 3, 5, 7, 11 and 13 are prime.

A composite number has more than two positive factors. For example, 12 has factors 1, 2, 3, 4, 6 and 12, so it is composite.

One is neither prime nor composite. It has only one distinct positive factor, not two. Two is the only even prime number: any even whole number greater than 2 is divisible by both 2 and another positive integer, so it cannot be prime.

A helpful question is, “Can we divide it exactly by any whole number other than 1 and itself?” A learner who answers without checking should explain a factor or a divisibility rule.

Prime factorisation is a building-block language

Every whole number greater than 1 can be expressed as a product of primes. Apart from the order of multiplication, those prime factors are uniquely determined.

For 36, start with 36 = 6 × 6. Since 6 = 2 × 3, we obtain 36 = 2 × 2 × 3 × 3 = 2² × 3².

For 60, we may start with 60 = 6 × 10. Expand the composite factors: 6 = 2 × 3 and 10 = 2 × 5. Therefore 60 = 2² × 3 × 5.

The factor tree can branch in different ways and still end with the same prime factors. A child who starts 60 = 3 × 20 should obtain the same final result. That is more important than drawing an identical tree to the tutor’s.

Check by multiplying the primes back together. For 60: 2 × 2 × 3 × 5 = 60. This simple check exposes missing factors before they cause errors in HCF or LCM.

Worked example 1: HCF of 36 and 60

Write 36 = 2² × 3² and 60 = 2² × 3 × 5.

For the HCF, keep only primes that occur in both factorizations, using the smaller number of copies of each shared prime. Both numbers have at least two factors of 2 and at least one factor of 3. The HCF is 2² × 3 = 12.

Check: 36 ÷ 12 = 3 and 60 ÷ 12 = 5, both whole numbers. Could a larger common factor exist? No. Any common factor needs only primes found in both numbers, and we have already selected every available shared prime factor.

This is what the exponent rule means. It is not simply “always choose the smaller exponent” without first asking which prime is common to both quantities.

Worked example 2: LCM of 36 and 60

Use the same factorizations: 36 = 2² × 3² and 60 = 2² × 3 × 5.

For the LCM, include every prime that appears in either number, taking enough copies to cover the larger exponent present. We need 2², 3² and 5, so LCM = 4 × 9 × 5 = 180.

Check: 180 ÷ 36 = 5 and 180 ÷ 60 = 3. Therefore 180 is a multiple of both. No smaller positive common multiple can omit any one of the required prime factors.

Notice a useful relationship for two positive integers: HCF × LCM = product of the two integers. Here 12 × 180 = 36 × 60 = 2160. This can check your calculation, although it should not replace understanding the original methods.

The same pair of numbers, but a different question

Imagine 36 blue cards and 60 orange cards. You want as many identical bundles as possible, with all cards used and each bundle having the same number of blue and orange cards. The greatest number of bundles is 12. Each bundle has 3 blue cards and 5 orange cards.

Now imagine a bell rings every 36 minutes and a second bell every 60 minutes, beginning together. Their next simultaneous ring is after 180 minutes. That is the LCM problem.

The quantities are identical, but the requested structure differs. In the first story we divide a total into equal groups. In the second we wait for cycles to align. Ask students to name the story before naming HCF or LCM.

Worked example 3: common factors without a factor tree

Find the HCF and LCM of 18 and 24 using lists.

Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. The greatest shared factor is 6.

Multiples of 18: 18, 36, 54, 72, 90, and so on. Multiples of 24: 24, 48, 72, 96, and so on. The first positive shared multiple is 72.

This list method is helpful with smaller numbers because it illustrates the definitions directly. For larger numbers, prime factorisation usually scales better and reduces the need to write many multiples.

Check: 18 × 24 = 432, and 6 × 72 = 432. The two approaches agree.

Worked example 4: three numbers

Find the HCF and LCM of 12, 18 and 30.

Prime factorisations: 12 = 2² × 3, 18 = 2 × 3², and 30 = 2 × 3 × 5.

For HCF, the shared primes are 2 and 3, each appearing at least once in all three numbers. HCF = 2 × 3 = 6.

For LCM, include the largest powers needed: 2² × 3² × 5 = 180. It is divisible by 12, 18 and 30.

One important warning: the shortcut HCF × LCM = product of the numbers is a relationship for two positive integers, not a general formula for three or more numbers. For 12, 18 and 30, 6 × 180 = 1080, while 12 × 18 × 30 = 6480. Avoid blindly extending a rule beyond its conditions.

How HCF helps simplify fractions

Simplify 48/60. The HCF of 48 and 60 is 12. Divide numerator and denominator by 12 to obtain 4/5.

Because the same nonzero factor divides both, the fraction’s value is unchanged. The two forms are equivalent: 48 ÷ 60 = 0.8, and 4 ÷ 5 = 0.8.

This links number theory to the Secondary 1 G3 ratio and sharing guide. Ratios and fractions become easier when a learner recognises the structure beneath the numbers.

Another example: 42/56 reduces by the HCF 14 to 3/4. If a child cancels mismatched digits instead of common factors, the apparent simplification may change the fraction’s value.

Square numbers and square roots by prime factorisation

A perfect square has each prime factor occurring an even number of times. Consider 144 = 2⁴ × 3². Pair identical prime factors: the square root contains 2² × 3 = 12. Hence √144 = 12, the principal nonnegative square root.

Check: 12 × 12 = 144. The prime factorisation explains why the square root is a whole number.

For 225 = 3² × 5², √225 = 3 × 5 = 15. Again, identical pairs are the clue.

A common confusion is to treat √49 as ±7. The symbol √49 denotes the principal nonnegative square root, 7. The equation x² = 49 has two solutions, x = 7 and x = −7. The statement being asked matters.

Not every square root is a whole number. For example, √50 is not an integer because 50 = 2 × 5² has an unpaired prime factor of 2.

Cube numbers and cube roots by prime factorisation

Cube roots group identical prime factors in threes. Since 216 = 2³ × 3³, its cube root is 2 × 3 = 6. Check: 6³ = 216.

Another example: 512 = 2⁹. Group the nine 2s into three groups of three. The cube root is 2³ = 8, and 8³ = 512.

For negative cubes, the real cube root can be negative. For instance, (−4)³ = −64, so the real cube root of −64 is −4. This differs from the usual principal square-root convention for negative numbers in real arithmetic.

A learner should describe what the grouping represents: pairs for square roots and groups of three for cube roots. Merely halving or dividing exponents without understanding the factor structure can become risky in more complicated settings.

Three misconceptions worth catching early

First, “1 is prime.” It is not. Primes have exactly two distinct positive factors, and 1 has one.

Second, “LCM is always the product of the two numbers.” That works for coprime positive integers, such as 8 and 15, which have HCF 1. It fails for 12 and 18: their product is 216, but the LCM is 36.

Third, “HCF is always the smaller number.” Sometimes it is, such as HCF(6, 24) = 6, but HCF(12, 18) = 6, which is smaller than both. The HCF follows divisibility, not numerical order alone.

For each misconception, ask for a counterexample. A student who can explain why a tempting rule fails is less likely to repeat it in an unfamiliar question.

Ten quick practice questions

• A. Is 1 a prime number? Explain.
• B. Prime-factorise 84.
• C. Find the HCF of 24 and 40.
• D. Find the LCM of 24 and 40.
• E. Simplify 42/56 fully.
• F. Find √196 using prime factors.
• G. Find the cube root of 216.
• H. Two timers repeat every 8 and 12 minutes. After how many minutes will they next coincide?
• I. Find the greatest number of identical bags possible using 36 red beads and 48 blue beads, with no leftovers.
• J. Prime-factorise 90, then explain whether it is a perfect square.

Answers with the reason included

A. No; 1 has exactly one positive factor.

B. 84 = 2² × 3 × 7. Check by multiplication: 4 × 3 × 7 = 84.

C. 24 = 2³ × 3 and 40 = 2³ × 5. HCF = 2³ = 8.

D. Include 2³, 3 and 5. LCM = 8 × 3 × 5 = 120.

E. HCF(42, 56) = 14, so 42/56 = 3/4.

F. 196 = 2² × 7², so √196 = 2 × 7 = 14.

G. 216 = 2³ × 3³, so its cube root is 6.

H. LCM(8, 12) = 24 minutes. Both cycles are complete then.

I. HCF(36, 48) = 12 identical bags. Each contains 3 red and 4 blue beads.

J. 90 = 2 × 3² × 5, with unpaired primes 2 and 5. It is not a perfect square.

If an answer is wrong, ask where the first unsupported decision occurred. Was the factorisation incomplete? Was HCF selected when the story needed LCM? Or was the question asking for the number of bags rather than the contents of each bag?

A ten-minute revision routine

Spend two minutes sorting six numbers into prime and composite. Then use three minutes to factorise two composite numbers and check each factor product.

Spend the next three minutes comparing a packaging problem and a recurring-events problem. Ask the learner to explain why one needs the HCF and the other the LCM before doing calculations. Use the last two minutes for a square-root or cube-root factorisation question.

A changed question later in the week is useful. Replace 36 and 60 with 24 and 40, or change an HCF grouping story into an LCM cycle story. If the student still identifies the structure correctly, the method is becoming transferable.

What parents can ask a Bukit Timah Mathematics tutor

Does the tutor check that the child understands factors, multiples and divisibility before introducing the exponent method? Can the learner explain why the smaller prime exponent helps find an HCF and the larger one helps find an LCM?

The eduKateSG Secondary 1 Mathematics small-group tutorial reference discusses close feedback and foundation repair in three-student classes near Sixth Avenue MRT. See also the Bukit Timah G3 Mathematics tuition overview for the wider programme context. Parents should confirm current availability and a student’s actual school topic sequence directly.

The official MOE G2 and G3 Mathematics syllabuses include primes, prime factorisation, HCF, LCM, and squares, cubes, square roots and cube roots by prime factorisation at Secondary 1 G3 level.

Frequently asked questions

Is prime factorisation really necessary if a calculator can find factors?

A calculator helps with arithmetic, but the purpose here is to see number structure. A student should still explain which primes combine to form a number and how those factors support HCF and LCM.

Is the HCF always smaller than the LCM?

For positive integers, the HCF is at most the smaller number and the LCM is at least the larger number. They can be equal, for example when both given numbers are 12.

Can I use repeated division instead of a factor tree?

Yes. A division ladder and factor tree both lead to the same prime factorisation when completed correctly. Choose a method the learner can explain and check.

Which topic should come next?

Ratio simplification, fractions and later algebra benefit from factor understanding. The significant figures and estimation guide extends number sense in a different direction.

Does G3 mean every child should do the hardest extension immediately?

No. G3 refers to the subject level under Full Subject-Based Banding. Build secure understanding of the current curriculum, then increase challenge when the child can solve changed problems without being coached.

Related Secondary 1 G3 Mathematics guides

• Significant figures, decimal places and estimation
• Number patterns and finding the nth-term rule
• Ratio, fractions and sharing quantities
• Algebraic expressions and bracket control

The best final question is not “Do you remember the rule?” It is “What is this problem asking us to find, and why is that a factor or a multiple?” When the child can answer, the method has a reason.