Question

A number $m$ is randomly selected from the set $\{11,13,15,17,19\},$ and a number $n$ is randomly selected from $\{1999,2000,2001,\ldots,2018\}.$ What is the probability that $m^n$ has a units digit of $1?$

$\dfrac{1}{5}$

$\dfrac{1}{4}$

$\dfrac{3}{10}$

$\dfrac{7}{20}$

$\dfrac{2}{5}$

Answer: E

Concepts:units digit modular exponentiation basic probability casework

Difficulty rating: 1540

Solution:

Only the units digit of $m$ matters. Among the $20$ consecutive possible exponents, each residue modulo $4$ occurs $5$ times, and $10$ exponents are even. A base ending in $1$ succeeds for all $20$ exponents; bases ending in $3$ or $7$ succeed for the $5$ exponents divisible by $4;$ a base ending in $5$ never succeeds; and a base ending in $9$ succeeds for the $10$ even exponents.

Thus $20+5+0+5+10=40$ of the $5\cdot20=100$ equally likely pairs work, and the probability is $\dfrac{40}{100}=\dfrac25.$

Thus, E is the correct answer.

2018 AMC 10A Problem 19

Answer: E

Difficulty rating: 1540

Solution:

Problem 19 in Other Years