Question

.
A radio station is hosting a telethon. Every 24th person to call in to the radio station will win $240, and every 11th person will win $111.
What is the number of the first caller to win both $240 and $111?

caller 241

caller 264

caller 288

caller 168

Answer 1

To find the first caller to win both $240 (for every 24th caller) and $111 (for every 11th caller), we need to determine the least common multiple (LCM) of the two numbers, 24 and 11.

1. **Determine the prime factorization**:
- For 24: \( 24 = 2^3 \times 3^1 \)
- For 11: \( 11 = 11^1 \) (11 is already prime)

2. **Calculate the LCM**:
To find the LCM, we take the highest powers of all prime factors involved:
- From 24: \( 2^3 \) and \( 3^1 \)
- From 11: \( 11^1 \)

So, the LCM is:
\[
\text{LCM} = 2^3 \times 3^1 \times 11^1
\]

3. **Calculate the LCM step-by-step**:
- First calculate \( 2^3 = 8 \)
- Next, calculate \( 8 \times 3 = 24 \)
- Finally, calculate \( 24 \times 11 = 264 \)

Thus, the LCM of 24 and 11 is 264.

4. **Conclusion**:
The first caller to win both $240 and $111 is caller number **264**.