Question

The number of integers between 1 and 500 (both inclusive) that are divisible by 3 or 5 or 7 is .

• A. 271

Answer 1

The correct option is A 271

Let A = {Integers between 1 to 500 divisible by 3}
B = {Integers between 1 to 500 divisible by 5}
C = {Integers between 1 to 500 divisible by 7}

To find number of integers between 1 to 500 that are divisible by 3 or 5 or 7 is

$n(A\cup B\cup C)=n\left(A\right)+n\left(B\right)+n\left(C\right)-n(A\cap B)$
$-n(A\cap C)-n(B\cap C)+n(A\cap B\cap C)$

$=(\left[\frac{500}{3}\right]+\left[\frac{500}{5}\right]+\left[\frac{500}{7}\right])-\{\left[\frac{500}{15}\right]+\left[\frac{500}{21}\right]+\left[\frac{500}{35}\right]\}+\left[\frac{500}{105}\right]$

$=(166+100+71)-(33+23+14)+4$
$=337-70+4=271$