Question

If the integers $m$ and $n$ are chosen at random between $1$ and $100$, then the probability that a number of the form $7^m+7^n$ is divisible by $5$ equals

• A. $1/5$
• B. $1/7$
• C. $1/8$
• D. None of these

Answer 1

The correct option is D None of these

We know $7^k,k\in N,$ has 1,3,9,7 at the unit's place for $k=4_p,4_p-1,4_p-2,4_p-3$ respectively, where p$=$ 1,2,3...
Clearly, $7^m+7^n$ will be divisible by 5 if $7^m$ has 3 or 7 in the unit's place and $7^n$ has 7 or 3 in the units place of $7^m$ has 1 or 9 in the unit's place and $7^n$ has 9 or 1 in the unit's place.
$\therefore$ For any choice of m,n the digit in the unit's place of $7^m+7^n$ is 2,4,6,0 or 8. It is divisible by 5 only when this digit is 0.
$\therefore$ the required probability $=\frac{1}{4}$