Question

Find the number of positive integers from 1 to 1000, which are divisible by atleast 2,3 or 5.

Answer 1

Let A be the no. of numbers divisible by 2.

B=no. of numbers divisible by 3.

C=no. of numbers divisible by 5.

$A=\left[\frac{1000}{2}\right]=500$

$B=\left[\frac{1000}{3}\right]=333$

$C=\left[\frac{1000}{5}\right]=200$

$A\cap B=\frac{1000}{6}]=166$

$B\cap C=\left[\frac{1000}{15}\right]=66$

$A\cap C=\left[\frac{1000}{10}\right]=100$

$A\cap B\cap C=\left[\frac{1000}{30}\right]=33$

$\therefore \cap (A\cup B\cup C)=500+333+200-166-66-100+33=734$