Question

The least common multiple of 20, 24, and 45 is _____.

180
360
21,600
30

Answer 1

To find the least common multiple (LCM) of the numbers 20, 24, and 45, we first find the prime factorization of each number:

- **20** can be factored into prime factors as \( 2^2 \times 5^1 \).
- **24** can be factored as \( 2^3 \times 3^1 \).
- **45** can be factored as \( 3^2 \times 5^1 \).

Next, to find the LCM, we take the highest power of each prime number that appears in the factorizations:

- For the prime number \( 2 \): The highest power is \( 2^3 \) (from 24).
- For the prime number \( 3 \): The highest power is \( 3^2 \) (from 45).
- For the prime number \( 5 \): The highest power is \( 5^1 \) (from both 20 and 45).

Now, we can calculate the LCM by multiplying these together:

\[
\text{LCM} = 2^3 \times 3^2 \times 5^1
\]

Calculating each part:

\[
2^3 = 8,
\]
\[
3^2 = 9,
\]
\[
5^1 = 5.
\]

Now we multiply these values together:

\[
8 \times 9 = 72,
\]
\[
72 \times 5 = 360.
\]

Thus, the least common multiple of 20, 24, and 45 is \( \boxed{360} \).