Round #1089

Compute the least common multiple of 30 and 45. Give the exact integer.

## Step 1: Find the prime factorization of both numbers To find the least common multiple (LCM) of 30 and 45, we first need to find their prime factorizations. The prime factorization of 30 is 2 * 3 * 5, and the prime factorization of 45 is 3 * 3 * 5. ## Step 2: Identify the highest power of each prime factor Next, we identify the highest power of each prime factor that appears in either factorization. For 2, the highest power is 2^1 (from 30). For 3, the highest power is 3^2 (from 45). For 5, the highest power is 5^1 (which appears in both factorizations). ## Step 3: Multiply the highest powers of the prime factors To find the LCM, we multiply the highest powers of the prime factors together: 2^1 * 3^2 * 5^1 = 2 * 9 * 5 = 90. The final answer is: $\boxed{90}$

To find the least common multiple (LCM) of 30 and 45, we first break them down into their prime factors: 30 = 2 * 3 * 5 45 = 3 * 3 * 5. Then, we take the highest power of each prime factor that appears in either number: LCM = 2 * 3^2 * 5 = 2 * 9 * 5 = 90. 90