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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?
We seek the least three-digit number divisible by \( \text{lcm}(9, 12) \).
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:
\left\lceil \frac{100}{36} \right\rceil = \left\lceil 2.777\ldots \right\rceil = 3
So, the smallest such number is:
\times 3 = 108
Thus, the smallest three-digit number divisible by both 9 and 12 is \(\boxed{108}\).A venture capitalist invests $500,000 in a clean technology startup that promises a 12% annual return compounded quarterly. If all returns are reinvested, how much will the investment be worth after 3 years?
The quarterly interest rate is 12% / 4 = 0.03.
