Calculate the smallest integer which is a multiple of 3, 5, and 10.

The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that each of the numbers divides into exactly. One way to find it is by prime factorization: express every number as a product of primes, then take each distinct prime factor to the highest power it appears in any of the factorizations, and multiply these together.

1. Prime factorize each number: 3 = 3, 5 = 5, 10 = 2 × 5.

2. Collect every distinct prime that appears across the factorizations: 2, 3, and 5. Each appears only once (to the first power) in any factorization.

3. Multiply these primes together: 2 × 3 × 5 = 30.

Check: 30 divided by 3 is 10, 30 divided by 5 is 6, and 30 divided by 10 is 3 — all divide evenly, confirming 30 is a common multiple. No smaller positive integer satisfies all three divisibility conditions.

Hence, the smallest integer that is a multiple of 3, 5, and 10 is 30.