Track / Stage 3 / 19 of 260 #19 of 2000

Problem 19

AMC 10/12, early questions

Number theory Difficulty 3.1 Find the answer

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

Your answer

Official 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 $\fbox{C}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.