Question

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

• A. $\frac{1}{4}$
• B. $\frac{1}{7}$
• C. $\frac{1}{8}$
• D. $\frac{1}{49}$

Answer 1

The correct option is A $\frac{1}{4}$

Number of possible cases $=100\times 100$
For $7^m+7^n$ is to be divisible by $5$:
$m$ ends with $9$ and $n$ ends with $1$
$\therefore m=2,6,10,...,98$ $n=4,8,12,...,100$
(25 cases) (25 cases)
$m$ ends with $7$ and $n$ ends with $3$
$\therefore m=1,5,9,13,...,97$ $n=1,5,9,13,...,97$
(25 cases) (25 cases)
$\because m$ and $n$ can be interchanged
$\therefore$ Required probability $=\frac{2\times 2\times 25\times 25}{100\times 100}=\frac{1}{4}$