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.
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Determine the prime factorization:
- For 24: \( 24 = 2^3 \times 3^1 \)
- For 11: \( 11 = 11^1 \) (11 is already prime)
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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: \[ \text{LCM} = 2^3 \times 3^1 \times 11^1 \]
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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.
- Conclusion: The first caller to win both $240 and $111 is caller number 264.
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