# How many of the integers 1, 2 - Quantitative Aptitude - CAT Equation

## CAT 2020 - Slot 3 - Quantitative Aptitude - Equation - Question 7 - How many of the integers 1, 2

Q. 7: How many of the integers 1, 2, … , 120, are divisible by none of 2, 5 and 7?

##### 1. 412. 423. 404. 43

We are aske dto find the number of numbers that are divisible by none of 2, 5 and 7. Instead, let us find the number of numbers that are divisible by atleast one of 2, 5 and 7.

Recall that n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C).

Therefore, Number of Multiples of (2 or 5 or 7) = Number of Multiples of (2) + Number of Multiples of (5) + Number of Multiples of (7) – Number of Multiples of (2 and 5) – Number of Multiples of (5 and 7) – Number of Multiples of (7 and 2) + Number of Multiples of (2 and 5 and 7)

Number of Multiples of (2 or 5 or 7) = Number of Multiples of (2) + Number of Multiples of (5) + Number of Multiples of (7) – Number of Multiples of (10) – Number of Multiples of (35) – Number of Multiples of (14) + Number of Multiples of (70)

Number of Multiples of (2 or 5 or 7) = 60 + 24 + 17 – 12 – 3 – 8 + 1 = 79

Since we know the number of integers that can be divided by atleast one of 2, 5 and 7 is 79. We can arrive at the number of integers that can be divided by none of 2, 5 and 7, by subtracting from the whole of 120.

No of integers of 1, 2, … , 120, that are divisible by none of 2, 5 and 7 = 120 – 79 = 41

### Past Year Question Paper & Solutions

Counselling Session
By IIM Mentor