Question

The sum of all the integers from $1$ to $100$ which are divisible by $2$ or $5$ is

• A. $3000$
• B. $3050$
• C. $3600$
• D. $3250$

Answer 1

The correct option is B $3050$

The A.P of the numbers divisible by $2$ between $1$ and $100$ is $2,\mathrm{4.......100}$.

Therefore, $a=2$ and $a_n=100$

$n=\frac{a_n-a}{d}+1$

$=\frac{100-2}{2}+1$

$=50$

Hence, $S_n=\frac{n}{2}(a+a_n)$

$=\frac{50}{2}\left(102\right)$

$=50\left(51\right)$

$=2550$ ........(i)

The A.P of the numbers divisible by $5$ between $1$ and $100$ is $5,\mathrm{10..100}$

$n=\frac{a_n-a}{d}+1$

$=\frac{100-5}{5}+1$

$=20$

Hence, $S_n=\frac{20}{2}105$

$=10\left(105\right)$

$=1050$ .......(ii)

The A.P of the numbers divisible by both $2$ and $5$ both is

$10,\mathrm{20..100}$

Therefore, $n=10$.

$S_n=10\left(\frac{110}{2}\right)$ $=550$ .......(iii)

Hence, the required sum is

$2550+1050-550$
$=3600-550$
$=3050$