Question

Find the number of positive integers less than 601 that are not divisible by 4 or 5 or 6

Answer 1

The number of positive integers less than 601 that are divisible by 4 

$$\lbrack\dfrac{600}{4}\rbrack=150$$

The number of positive integers less than 601 that are divisible by 6 

$$\lbrack\dfrac{600}{6}\rbrack=100$$

The number of positive integers less than 601 that are divisible by 5 

$$\lbrack\dfrac{600}{5}\rbrack=120$$

The number of positive integers less than 601 that are divisible by 4 and 6

$$\lbrack\dfrac{600}{12}\rbrack=50$$

The number of positive integers less than 601 that are divisible by 4 and 5 

$$\lbrack\dfrac{600}{20}\rbrack=30$$

The number of positive integers less than 601 that are divisible by 5 and 6 

$$\lbrack\dfrac{600}{30}\rbrack=20$$

The number of positive integers less than 601 that are divisible by 4, and 5, and 6 

$$\lbrack\dfrac{600}{60}\rbrack=10$$

The number of positive integers less than 601 that are divisible by 4 or 5 or 6

$$150+100+120-50-30-20+10=280$$

The number of positive integers less than 601 that are not divisible by 4 or 5 or 6

$$600-280=320$$