Discussion
What is the remainder when \(2^{2024}\) is divided by 15?
Answer: A
We use Euler's Totient Theorem. First, find \(\phi(15) = \phi(3 \times 5) = \phi(3)\phi(5) = (3-1)(5-1) = 2 \times 4 = 8\). According to the theorem, \(2^8 \equiv 1 \pmod{15}\). We need to find the remainder of the power \(2024\) when divided by 8. The rule for divisibility by 8 is to check the last three digits. \(024\) is divisible by 8. So, \(2024\) is a multiple of 8. We can write \(2024 = 8k\). The expression becomes \((2^8)^k \equiv 1^k \equiv 1 \pmod{15}\). The remainder is 1.
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