Question

If the integers $a$ and $b$ are chosen randomly between $1$ to $100$, then the probability that a number of the form $7^a+7^b$ is divisible by $5$ is

• A. $\frac{1}{7}$
• B. $\frac{1}{5}$
• C. $\frac{1}{48}$
• D. $\frac{1}{4}$

Answer 1

The correct option is B $\frac{1}{5}$

Fact $7^m,mϵN$ has $1,3,9,7$ at the unit place for $m=4p,4p-1,4p-2,4p-3\forall a=1,2,3\cdots$
So $7^a+7^b$ will be divisible by $5$ if $7^a$ has $3$ or $7$ at unit place and $7^a$ has $7$ or $3$ at unit place or $7^a$ has $4$ or $9$ at the unit place and $7^a$ has $9$ ro $1$ at the unit place.
$\therefore$ for choice of a and b the unit place in the expression $7^a+7^b$ is $0,2,4,6,8$ and will be divisible by $5$ only if $0$ occurs at the unit place.
$\therefore$ Required probability $=\frac{1}{5}$