Question

How many integers between $50$ and $500$ which are divisible by $7$?

Answer 1

We observe that $56$ is the first integer between $50$ and $500$ which is divisible by $7$.

Also, when we divide $500$ by $7$ the remainder is $3$. Therefore, $500-3=497$ is the largest integer divisible by $7$ and lying between $50$ and $500$.

Thus, we have to find the number of terms in an A.P. with first term $=56$, last term $=497$ and common difference $=7$ (as the numbers are divisible by $7$).

Let there be $n$ terms in the A.P. Then,
$a_n=497$

$⟹a+(n-1)d=497$
$⟹56+(n-1)\times 7=497$ $[\because a=56andd=7]$
$⟹7n+49=497$
$⟹7n=448⟹n=64$
Thus. there are $64$ integers between $50$ and $500$ which are divisible by $7$.