HCF and LCM Questions with Answers: Methods and 10 MCQs

HCF (highest common factor) and LCM (least common multiple) questions appear in JKSSB Class-IV and Junior Assistant papers and in SSC CGL. A few rules cover almost all of them.

Finding HCF and LCM

• Prime factorisation: write each number as a product of primes. HCF takes the lowest power of each common prime; LCM takes the highest power of every prime\n- Example: 24 = 2³ × 3 and 36 = 2² × 3². HCF = 2² × 3 = 12; LCM = 2³ × 3² = 72\n- Check: 12 × 72 = 864 = 24 × 36

The four problem types

• Find the other number: other = (HCF × LCM) ÷ the given number\n- Together again: bells, lights or buses that repeat at different intervals meet again after the LCM of the intervals\n- Smallest number divisible by several numbers: the LCM\n- Divides leaving remainders: subtract each remainder, then take the HCF; for 'smallest number leaving remainder r', take LCM + r

More worked examples

• Remainder problem: find the largest number that divides 245 and 1,029 and leaves remainder 5 in each case. Subtract the remainder to get 240 and 1,024. The HCF of 240 and 1,024 is 16
• Smallest number with remainders: the smallest number that leaves remainder 3 when divided by 4, 5 and 6 is LCM(4, 5, 6) + 3 = 60 + 3 = 63
• Bells: three bells ring every 12, 15 and 20 minutes. They next ring together after LCM(12, 15, 20) = 60 minutes. If they ring together at 8:00, they ring together again at 9:00
• Ratio and HCF: two numbers are in the ratio 3 : 4 and their HCF is 5. The numbers are 15 and 20, and their LCM is 60
• Tiles: a floor is 6.60 m by 4.80 m. The largest square tile that fits exactly has a side equal to the HCF of 660 cm and 480 cm, which is 60 cm. The number of tiles is 11 × 8 = 88

HCF and LCM of fractions

For fractions in their simplest form, the HCF is the HCF of the numerators divided by the LCM of the denominators, and the LCM is the LCM of the numerators divided by the HCF of the denominators.

Example: find the HCF of 2/3, 4/9 and 8/27. The HCF of the numerators 2, 4 and 8 is 2. The LCM of the denominators 3, 9 and 27 is 27. So the HCF is 2/27. The LCM of the same fractions is LCM(2, 4, 8) ÷ HCF(3, 9, 27) = 8 ÷ 3 = 8/3.

Co-prime numbers and special cases

Two numbers are co-prime when their HCF is 1. For co-prime numbers, the LCM is simply their product. For example, 8 and 15 are co-prime, so their LCM is 120. If one number is a multiple of the other, the smaller is the HCF and the larger is the LCM. For 6 and 18, the HCF is 6 and the LCM is 18.

The division method for HCF

When the numbers are large, prime factorisation is slow. Divide the larger number by the smaller one, then divide the divisor by the remainder, and repeat until the remainder is zero. The last divisor is the HCF. For 252 and 105: 252 ÷ 105 leaves 42, 105 ÷ 42 leaves 21, and 42 ÷ 21 leaves 0, so the HCF is 21. Then the LCM is 252 × 105 ÷ 21 = 1,260.

Common mistakes

• Mixing up the two. HCF is never larger than the smallest number, and LCM is never smaller than the largest number
• Taking the highest power for HCF and the lowest for LCM, which is the reverse of the rule
• Forgetting to subtract the remainders before finding the HCF in a remainder problem
• Adding the remainder in the wrong problem type: for the smallest number that leaves the same remainder, add it to the LCM

Practise

Try the 10 MCQs below, then continue with HCF & LCM practice sets for Class-IV and Junior Assistant. For related methods see the Class-IV maths guide.