We seek the least three-digit number divisible by \( \text{lcm}(9, 12) \).
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- \gcd(2^{10} - 1, 2^{15} - 1) = 2^5 - 1 = 32 - 1 = 31
- The greatest common divisor is \(\boxed{31}\).
- Question:** A zoologist tracking nocturnal animal activity cycles notices behavioral patterns recurring every 9 days, prompting interest in numerical periodicity. What is the smallest three-digit number that is divisible by both 9 and 12?
- First, compute the LCM:
- = 3^2, \quad 12 = 2^2 \times 3 \quad \Rightarrow \quad \text{lcm}(9,12) = 2^2 \times 3^2 = 36
- Now, find the smallest three-digit multiple of 36:
