1967 AHSME Problems/Problem 25

Problem

For every odd number $p>1$ we have:

$\textbf{(A)}\ (p-1)^{\frac{1}{2}(p-1)}-1 \; \text{is divisible by} \; p-2\qquad \textbf{(B)}\ (p-1)^{\frac{1}{2}(p-1)}+1 \; \text{is divisible by} \; p\\ \textbf{(C)}\ (p-1)^{\frac{1}{2}(p-1)} \; \text{is divisible by} \; p\qquad \textbf{(D)}\ (p-1)^{\frac{1}{2}(p-1)}+1 \; \text{is divisible by} \; p+1\\ \textbf{(E)}\ (p-1)^{\frac{1}{2}(p-1)}-1 \; \text{is divisible by} \; p-1$

Solution

$\fbox{A}$ Given that $p$ is odd, $p-1$ must be even, therefore ${{\frac{1}{2}}(p-1)}$ must be an integer, which will be denoted as n. $(p-1)^n-1$ By sum and difference of powers $=((p-1)-1)((p-1)^{n-1}+\cdots+1^{n-1})$ $=(p-2)((p-1)^{n-1}+\cdots+1^{n-1})$ Therefore $p-2$ divide $(p-1)^{{\frac{1}{2}}(p-1)}$

1967 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 24	Followed by Problem 26
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1967 AHSME Problems/Problem 25

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