HCF and LCM for AFCAT: methods, the product rule and remainder questions

HCF and LCM questions in AFCAT are rarely hard, but they are easy to get wrong in one specific way: using HCF when the question needs LCM, or the other way round. The arithmetic is simple. The real skill is reading the question and deciding which one it wants, and then handling the remainder twist that examiners like to add.

This guide builds on the prime factorisation from the number system guide. It covers the two methods, the product rule and why it holds, HCF and LCM of fractions, and each type of remainder question.

What they mean

• HCF (highest common factor) is the largest number that divides all the given numbers exactly. It is always ≤ the smallest number.
• LCM (lowest common multiple) is the smallest number that all the given numbers divide exactly. It is always ≥ the largest number.
• The HCF always divides the LCM.

Method 1: prime factorisation

Write each number as a product of prime powers.

• HCF = product of the common primes, each to its lowest power.
• LCM = product of all the primes that appear, each to its highest power.

For 36 = 2² × 3² and 48 = 2⁴ × 3:

• HCF = 2² × 3 = 12.
• LCM = 2⁴ × 3² = 144.

Why: a common factor can use a prime only as many times as the number with fewest copies allows. A common multiple must contain every prime as many times as the number with most copies needs.

Method 2: the division method for HCF

For large numbers that are hard to factorise, divide the larger by the smaller, then divide the divisor by the remainder, and repeat until the remainder is 0. The last divisor is the HCF.

Why it works: any number that divides both a and b also divides a − b, and so divides the remainder when a is divided by b. The HCF never changes as the numbers shrink.

The product rule

For two numbers: HCF × LCM = product of the two numbers.

Why: for each prime, the HCF takes the smaller power and the LCM the larger. Between them they use both powers exactly once, which is what the product of the two numbers does.

This rule does not hold for three or more numbers.

Also useful: if two numbers have HCF h, they can be written as h × a and h × b, where a and b have no common factor. Their LCM is then h × a × b.

HCF and LCM of fractions

• HCF of fractions = HCF of the numerators ÷ LCM of the denominators.
• LCM of fractions = LCM of the numerators ÷ HCF of the denominators.

Write each fraction in its lowest terms first.

Which one does the question want?

Worked examples

Example 1 (division method). Find the HCF of 391 and 667.

• 667 = 1 × 391 + 276.
• 391 = 1 × 276 + 115.
• 276 = 2 × 115 + 46.
• 115 = 2 × 46 + 23.
• 46 = 2 × 23 + 0, so the HCF is 23.
• Check: 391 = 17 × 23 and 667 = 29 × 23.

Example 2 (product rule). The HCF of two numbers is 6 and their LCM is 180. One number is 30. Find the other.

• Other number = 6 × 180 ÷ 30 = 36.
• Check: 30 = 2 × 3 × 5 and 36 = 2² × 3². HCF = 6 and LCM = 2² × 3² × 5 = 180.

Example 3 (ratio and LCM). Two numbers are in the ratio 3 : 4 and their LCM is 180. Find the numbers.

• Write them as 3h and 4h, where h is the HCF. Their LCM is 3 × 4 × h = 12h.
• 12h = 180, so h = 15. The numbers are 45 and 60.

Example 4 (same unknown remainder). Find the largest number that divides 62, 132 and 237, leaving the same remainder in each case.

• If all three leave the same remainder, the differences are exact multiples of the divisor.
• Differences: 132 − 62 = 70, 237 − 132 = 105, 237 − 62 = 175.
• HCF of 70, 105 and 175 = 35.
• Check: each number leaves remainder 27 when divided by 35.

Example 5 (constant gap). Find the smallest number that leaves remainders 8, 11 and 16 when divided by 12, 15 and 20.

• In each case the remainder is 4 less than the divisor (12 − 8 = 15 − 11 = 20 − 16 = 4).
• So the number is 4 short of a common multiple: LCM(12, 15, 20) − 4 = 60 − 4 = 56.
• Check: 56 = 4 × 12 + 8 = 3 × 15 + 11 = 2 × 20 + 16.

Example 6 (tiles). A floor measures 6 m 24 cm by 4 m 32 cm. It is to be covered with the largest possible square tiles, all the same size. Find the tile size and the number of tiles.

• In centimetres: 624 and 432. HCF = 48 cm.
• Tiles = (624 ÷ 48) × (432 ÷ 48) = 13 × 9 = 117.

Common mistakes

• Choosing the wrong one. "Largest" usually means HCF, and "smallest" or "together again" usually means LCM.
• Forgetting the remainder adjustment: add r for the smallest number with a fixed remainder; subtract r from each number for the largest divisor.
• Impossible pairs. An HCF must divide the LCM. A question that offers HCF 16 and LCM 100 has no such numbers, and an option built on it can be ruled out.
• Units in tile and rope questions. Convert metres to centimetres before finding the HCF.

Practice set

1. Find the HCF of 84 and 126.
2. Find the LCM of 12, 15 and 20.
3. Find the smallest number that leaves remainder 4 when divided by 6, 9 and 12.
4. Find the HCF and LCM of 2/3, 4/9 and 8/15.
5. Find the largest number that divides 43 and 91, leaving remainder 7 in each case.
6. The product of two numbers is 2,160 and their HCF is 12. Find their LCM.
7. Find the greatest four-digit number divisible by 12, 18 and 30.
8. Three runners complete a lap in 30, 45 and 60 seconds. They start together at the start line. After how long are they together there again?

Answers:

1. 42. 84 = 2² × 3 × 7 and 126 = 2 × 3² × 7, so HCF = 2 × 3 × 7.
2. 60. 2² × 3 × 5.
3. 40. LCM(6, 9, 12) = 36, and 36 + 4.
4. HCF 2/45, LCM 8/3. HCF(2, 4, 8) = 2 and LCM(3, 9, 15) = 45. LCM(2, 4, 8) = 8 and HCF(3, 9, 15) = 3.
5. 12. HCF(43 − 7, 91 − 7) = HCF(36, 84) = 12.
6. 180. 2,160 ÷ 12.
7. 9,900. LCM = 180. 9,999 ÷ 180 gives 55 with remainder 99, so 55 × 180 = 9,900.
8. 3 minutes. LCM(30, 45, 60) = 180 seconds.

What to do next

• Memorise the "which one" table, including the remainder adjustments.
• Practise ten remainder questions, checking each answer by actually dividing.
• Revise number system divisibility rules, then move on to approximation and the numerical ability plan to see where this topic fits.