The sum of the integers lying between 1 and 100 and divisible by 3 or 5 or 7 is

The correct option is A: $2738$

The integers divisible by 3 are 33 in numbers.
The integers divisible by 5 are 20 in numbers.
The integers divisible by 7 are 14 in numbers.
The integers divisible by both 3 and 5 are 6 in numbers.
The integers divisible by both 3 and 7 are 4 in numbers.
The integers divisible by both 5 and 7 are 2 in numbers.
There are no integers divisible by all three.
Hence the sum of numbers divisible by
3 or 5 or 7 is
$=\frac{33}{2}(3+99)+\frac{20}{2}(5+95)+\frac{14}{2}(7+98)-\frac{6}{2}(15+90)-\frac{4}{2}(21+84)-\frac{2}{2}(35+70)$

$=33\left(51\right)+1000+735-315-210-105$

$=1685+1000+735-630=2790$