Question

The least common multiple of 72 and 120

Answer 1

To find the least common multiple (LCM) of 72 and 120, we can use the prime factorization method.

First, we find the prime factorization of each number:

- **72**:
- \(72 = 8 \times 9 = 2^3 \times 3^2\)

- **120**:
- \(120 = 12 \times 10 = (3 \times 4) \times (2 \times 5) = 3 \times (2^2) \times 2 \times 5\)
- Simplifying: \(120 = 2^3 \times 3^1 \times 5^1\)

Now the prime factorizations are:
- \(72 = 2^3 \times 3^2\)
- \(120 = 2^3 \times 3^1 \times 5^1\)

To find the LCM, we take the highest power of each prime factor present in the factorizations:

- The highest power of \(2\) is \(2^3\).
- The highest power of \(3\) is \(3^2\).
- The highest power of \(5\) is \(5^1\).

Now, we can multiply these together to get the LCM:
\[
\text{LCM} = 2^3 \times 3^2 \times 5^1
\]

Calculating this step-by-step:
- \(2^3 = 8\)
- \(3^2 = 9\)
- \(5^1 = 5\)

Now, multiply these together:
\[
8 \times 9 = 72
\]
\[
72 \times 5 = 360
\]

Thus, the least common multiple of 72 and 120 is \(\boxed{360}\).