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What is the least common multiple (LCM) of the numbers 20, 30, and 40?
60
120
40
30
The correct answer is: 120
To determine the least common multiple (LCM) of 20, 30, and 40, we begin by finding the prime factorization of each number: - The prime factorization of 20 is \(2^2 \times 5\). - The prime factorization of 30 is \(2 \times 3 \times 5\). - The prime factorization of 40 is \(2^3 \times 5\). Next, we identify the highest power of each prime number that appears in these factorizations: - The highest power of 2 among the numbers is \(2^3\) (from 40). - The highest power of 3 is \(3^1\) (from 30). - The highest power of 5 is \(5^1\) (common to all). Combining these highest powers gives us the LCM: \[ LCM = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 \] Calculating this step-by-step: 1. \(8 \times 3 = 24\) 2. \(24 \times 5 = 120\)