2018 AMC 10A Problem 19

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.