1974 AHSME Problems/Problem 8

Problem

What is the smallest prime number dividing the sum $3^{11}+5^{13}$ ?

$\mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \ } 5 \qquad \mathrm{(D) \ } 3^{11}+5^{13} \qquad \mathrm{(E) \ }\text{none of these}$

Solution

Since we want to find the smallest prime dividing the sum, we start with the smallest prime and move up, so first we try $2$ . Notice that $3^{11}$ and $5^{13}$ are both odd, so their sum must be even. This means that $2$ must divide $3^{11}+5^{13}$ , and so since $2$ is the smallest prime, our answer must be $2, \boxed{\text{A}}$ .

1974 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
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1974 AHSME Problems/Problem 8

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