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What is the least common multiple (LCM) of the numbers 20, 30, and 40?

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. $24 \times 5 = 120$

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