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Number Theory · Topic 5 of 9

🔗 LCM-based Problems

1 exam question type, fully solved

Many word problems are really one LCM (or HCF) sum plus a small adjustment. Spot the pattern and the answer comes in two steps.

The three patterns

• Exactly divisible by all the divisors → answer = their LCM (or LCM × k).

• Same remainder r with each divisor → answer = LCM + r.

• Different remainders, but (divisor − remainder) is the same constant k → answer = LCM − k.

Extra condition? (e.g. “also divisible by 13”) — write the number as LCM × k + r and test k = 1, 2, 3 … until it fits.

📊Multiples GridSee multiples of two numbers highlighted on 1–100
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Lowest Common Multiple of 3 and 4 =12

Least number divisible by 12, 15 and 20 = their LCM = .

📝Practice Questions

Q1The least number divisible by 12, 15 and 20 is:

Q2The least number that leaves remainder 3 when divided by 5, 6 and 8 is:

Q3The least number which when divided by 7 and 9 leaves remainders 5 and 7 respectively is:

📚

Real exam questions — LCM Problems

1 question types · 4 solved examples from real SSC papers

These word problems all reduce to one LCM (or HCF) calculation plus a small adjustment.

How to solve this type
Least number exactly divisible by all the divisors = their LCM (take LCM × k for higher multiples). If each divisor leaves the SAME remainder r, the answer is LCM + r. If the remainders differ but (divisor − remainder) is the same constant k for all, the answer is LCM − k. When an extra condition is added (e.g. the number must also be divisible by 13), write the number as LCM × k + r and test k = 1, 2, 3 … until it holds. For "greatest number leaving the same remainder", take the HCF of the differences of the given numbers.

What is the least number which, when divided by 8, 12, 15, 24, 25 and 40, leaves remainder 7 in each case?

A597B627C607D617

Find the least number which, when divided by 15, 18 and 42, leaves remainder 8 in each case, and is also exactly divisible by 13. What is the sum of its digits?

A26B25C22D24

Find the least number which, when divided by 36, 72, 80 and 88, leaves remainders 16, 52, 60 and 68 respectively. What is the sum of its digits?

A17B11C14D16

Find the greatest number that divides 49, 147 and 322, leaving the same remainder in each case.

A7B21C49D14