Find the least number which, when divided by 12, 20, and 15, leaves no…

The Least Common Multiple (LCM) of a set of numbers is the smallest positive number that is exactly divisible by every number in the set.

It is found via prime factorization: express each number as a product of primes, then take the highest power of every prime factor that appears across all the numbers and multiply them together.

1. Write each number as a product of prime factors: 12 = 22 × 3, 20 = 22 × 5, 15 = 3 × 5.

2. Take the highest power of each prime factor that appears in any of the three numbers: 22 (from 12 and 20), 3¹ (from 12 and 15), and 5¹ (from 15 and 20).

3. Multiply these together: LCM = 22 × 3 × 5 = 4 × 3 × 5 = 60.

Cross-check: 60 ÷ 12 = 5, 60 ÷ 20 = 3, and 60 ÷ 15 = 4 — each division is exact, so 60 leaves no remainder with any of the three numbers.

Hence, the required least number is 60.