HCF and LCM

HCF (Highest Common Factor) = the largest number dividing all given numbers. LCM (Lowest Common Multiple) = smallest number divisible by all. KEY RELATION: for any two numbers a and b, HCF x LCM = a x b. This lets you find one when three values are known. HCF is always less than or equal to the smallest number; LCM is always greater than or equal to the largest number. HCF always divides LCM exactly. For fractions: HCF of fractions = HCF(numerators)/LCM(denominators); LCM of fractions = LCM(numerators)/HCF(denominators). Memory aid: 'Highest Cuts Fractions' (HCF uses HCF on top), opposite for LCM.

Prime factorisation method: write each number as a product of primes. HCF = product of the LOWEST powers of common primes. LCM = product of the HIGHEST powers of all primes appearing. Example: 12 = 2^2 x 3, 18 = 2 x 3^2. HCF = 2^1 x 3^1 = 6; LCM = 2^2 x 3^2 = 36. Check: 6 x 36 = 216 = 12 x 18. The division (ladder) method is faster for LCM of many numbers: divide repeatedly by common primes, then multiply all divisors and remaining quotients. For HCF of large numbers, the long-division (Euclidean) method is quickest.

BELLS RINGING TOGETHER: if bells toll at intervals of 6, 9, 12 seconds, they ring together after LCM(6,9,12) = 36 seconds; in 1 hour (3600 s) together = 3600/36 + 1 times (the +1 counts the start). COMMON REMAINDER: 'smallest number which when divided by a, b, c leaves the same remainder r' = LCM(a,b,c) + r. 'Largest number dividing x, y, z leaving remainder r each' = HCF(x-r, y-r, z-r), or HCF of the differences when remainders are equal but unknown. These two templates cover the majority of RPF SI HCF/LCM word problems — identify which one the question fits first.

Cover the answer, recall, then check. 12 cards on HCF/LCM shortcuts for RPF SI.

Q1. State the master relation between HCF and LCM of two numbers.
A1. HCF × LCM = product of the two numbers. So one unknown is found if the other three are known.

Q2. HCF of fractions formula?
A2. HCF = HCF(numerators) / LCM(denominators). E.g. HCF of 2/3, 4/9 = HCF(2,4)/LCM(3,9) = 2/9.

Q3. LCM of fractions formula?
A3. LCM = LCM(numerators) / HCF(denominators). E.g. LCM of 2/3, 4/9 = LCM(2,4)/HCF(3,9) = 4/3.

Q4. Largest number dividing 70 and 125 leaving remainders 5 and 8?
A4. HCF of (70−5)=65 and (125−8)=117 = 13.

Q5. Least number that when divided by 6, 7, 8 leaves remainder 3 each time?
A5. LCM(6,7,8) + 3 = 168 + 3 = 171.

Q6. HCF of two numbers is 8, LCM is 48. If one number is 24, find the other.
A6. Other = (8×48)/24 = 16.

Q7. For co-prime numbers, what is their HCF and LCM?
A7. HCF = 1; LCM = their product.

Q8. Least number leaving the SAME remainder r when divided by a, b, c?
A8. LCM(a,b,c) + r (for the smallest such number greater than the divisors' spread).

Q9. Number that divides a,b,c leaving remainders r1,r2,r3 respectively?
A9. HCF of (a−r1), (b−r2), (c−r3).

Q10. LCM of 12, 15, 20?
A10. 60 (2²×3×5).

Q11. HCF of 12, 15, 20?
A11. 1 — they share no common prime factor across all three.

Q12. Greatest number of 4 digits divisible by 12, 18, 21?
A12. LCM = 252; largest 4-digit multiple = 9999 − (9999 mod 252) = 9999 − 171 = 9828.

Worked Example

Problem: Two numbers are in the ratio 3 : 4 and their LCM is 240. Find the two numbers and their HCF, and verify the relation HCF × LCM = product of the two numbers.

Solution:
Step 1 — Represent the numbers using the ratio. Since the numbers are in ratio 3 : 4, write them as 3x and 4x, where x is their HCF (the common factor).
Step 2 — Use the LCM. For numbers 3x and 4x, LCM = 12x (because 3 and 4 are co-prime, LCM of 3x, 4x = 12x).
So 12x = 240 → x = 20.
Step 3 — Find the numbers: 3x = 3 × 20 = 60 and 4x = 4 × 20 = 80.
Step 4 — HCF = x = 20.
Step 5 — Verify: HCF × LCM = 20 × 240 = 4800, and product = 60 × 80 = 4800. Both match.

Answer: The numbers are 60 and 80, their HCF is 20, and HCF × LCM = 4800 = product.