CAT 2020 Question Paper | Quants Slot 3

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This question is from Number Theory. A range of integers is given. We have to find how many of them are not divisible by the other three numbers. A range of CAT questions can be asked based on this simple concept of divisibility. CAT exam has always tested the idea of divisibility and multiples from Number systems. Make use of 2IIMs CAT Question Bank, provided with detailed Video explanations to obtain a kick-ass CAT score.

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

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Video Explanation

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Explanatory Answer

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

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The question is "How many of the integers 1, 2, … , 120, are divisible by none of 2, 5 and 7?"

Hence, the answer is, "41"

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