Question

At a local high school, two copy machines were cleaned and serviced last week. Copy Machine A is cleaned and serviced every 12 weeks and Machine B is serviced every 8 weeks. How many more weeks will pass before they are both serviced again in the same week?

Answer 1

To determine how many more weeks will pass before both copy machines A and B are serviced again in the same week, we need to find the least common multiple (LCM) of their servicing intervals.

Copy Machine A is serviced every 12 weeks, and Copy Machine B is serviced every 8 weeks.

1. **Find the prime factorization of each number**:
- The prime factorization of \( 12 \) is \( 2^2 \times 3^1 \).
- The prime factorization of \( 8 \) is \( 2^3 \).

2. **Determine the LCM**:
The LCM is found by taking the highest power of each prime that appears in the factorizations:
- For the prime \( 2 \), the highest power is \( 2^3 \) (from 8).
- For the prime \( 3 \), the highest power is \( 3^1 \) (from 12).

Now calculate the LCM:
\[
\text{LCM} = 2^3 \times 3^1 = 8 \times 3 = 24
\]

3. **Calculate the time until both machines are serviced again**:
Both machines will be serviced together every 24 weeks.

Since they were both serviced last week (consider this week as week 0), we need to find out how many more weeks will pass until they are serviced together again.

\[
\text{Weeks until next servicing together} = 24 - 0 = 24 \text{ weeks}
\]

Thus, the answer is \(\boxed{24}\).