Number System & Simplification • Topic 3 of 6

HCF & LCM

HCF (greatest common divisor) is the largest number dividing all the given numbers; LCM (least common multiple) is the smallest number all of them divide. Use prime factorisation: HCF takes the lowest powers of common primes, LCM the highest powers of all primes. The identity HCF x LCM = product of the two numbers is a constant SSC favourite. Word problems about bells ringing together, tiles, or the largest measuring rod are LCM/HCF in disguise.

✅ Solved examples

1. Find the HCF and LCM of 24 and 36.
24 = 2^3 x 3, 36 = 2^2 x 3^2. HCF = 2^2 x 3 = 12. LCM = 2^3 x 3^2 = 72.
2. The HCF of two numbers is 8 and their LCM is 96. If one number is 32, find the other.
Product = HCF x LCM = 8 x 96 = 768. Other = 768/32 = 24.
3. Find the least number divisible by 12, 15 and 20.
LCM(12,15,20) = 60. Answer 60.
4. Three bells ring at intervals of 6, 8 and 12 seconds. After how many seconds do they ring together?
LCM(6,8,12) = 24 seconds.

✏️ Practice — try these, take hints as needed

1. HCF of 48 and 72?
48 = 2^4 x 3, 72 = 2^3 x 3^2.
Lowest powers.
2^3 x 3.
24
2. LCM of 9, 12 and 15?
Primes 3^2, 2^2 x 3, 3 x 5.
Highest powers.
2^2 x 3^2 x 5.
180
3. Two numbers are in ratio 3:4 and their LCM is 84. Find them.
Let 3x and 4x.
LCM = 12x = 84.
x = 7.
21 and 28
4. Largest number dividing 70 and 125 leaving remainders 5 and 8?
It divides 70-5 = 65 and 125-8 = 117.
HCF(65,117).
65 = 5 x 13, 117 = 9 x 13.
13
5. Least number leaving remainder 3 when divided by 5, 6 and 8?
LCM(5,6,8) = 120.
Add the remainder.
120 + 3.
123

📝 Topic test — 8 questions

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