Question

The number of ordered pairs (m,n) such that $(m,nϵ\{1,2,3,...20\})$ such that $3^m+7^n$ is a multiple of 10 is

• A. 200
• B. 100
• C. 16
• D. 40

Answer 1

The correct option is B 100

The number $3^m+7^n$ is divisible by $10$ only when the last digit of both add up to make $10$
In this case it is possible only when $3^m$ ends with $9$ and $7^n$ ends with $1$ or vice-versa.Also when $3^m$ ends with $3$ and $7^n$ ends with $7$ and vice-versa.
$3^m$ ends with $9$ when $m=2,6,10,14,18$ and $7^n$ ends with $1$ when $n=4,8,12,16,20$
Hence no. of ways the addition results $10=5\times 5=25$
Similarly $3^m$ ends with $1$ when $m=4,8,12,16,20$ and$7^n$ ends with $9$ when $n=2,6,10,14,18$
Hence no. of ways the addition results $10=5\times 5=25$
Similarly in other two cases also we can see that the addition results $10$ in $5\times 5=25$ ways.
Therefore, total number of $(m,n)$ pairs$=4\times 25=100$
Hence, option 'B' is correct.