1990 AHSME Problems/Problem 18

Problem

First $a$ is chosen at random from the set $\{1,2,3,\cdots,99,100\}$ , and then $b$ is chosen at random from the same set. The probability that the integer $3^a+7^b$ has units digit $8$ is

$\text{(A) } \frac{1}{16}\quad \text{(B) } \frac{1}{8}\quad \text{(C) } \frac{3}{16}\quad \text{(D) } \frac{1}{5}\quad \text{(E) } \frac{1}{4}$

Solution

The units digits of the powers of $3$ and $7$ both cycle through $1,3,9,7$ in opposite directions, and as $4\mid 100$ each power's units digit is equally probable. There are $16$ ordered pairs of units digits, and three of them $(1,7),(7,1),(9,9)$ have a sum with units digit $8$ .

Thus the probability is $\frac3{16}$ which is $\fbox{C}$

1990 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 17	Followed by Problem 19
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1990 AHSME Problems/Problem 18

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