Track / Stage 3 / 23 of 260 #23 of 1964

Problem 23

AMC 10/12, early questions

Number theory Difficulty 3.1 Multiple choice

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

Pick one

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 reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.