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## Q.58) The number of integers between 1 and 500 (both inclusive) that are divisible by 3 or 5 or 7 is ______. -gate computer science 2017

**A) 269**

**B) 270**

**C) 271**

**D) 272**

By:satyashiromani

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## Answers

**C) 271**

Explanation:

The general formula for the union of 3 sets is:

(A union B union C) = A + B + C - (A intersect B) - (A intersect C) - (B intersect C) + (A intersect B intersect C).

Assuming,

A = 3, B = 5, C = 7

= 500/3 + 500/5 + 500/7 - 500/3*5 - 500/5*7 - 500/7*3 + 500/105

= 271

Therefore, option C is correct.

Alternate Solution: Number of integers divisible by 3 or 5 or 7 = n (3 V 5 V 7) = n (3) + n(5) + n (7) - n (3 \wedge 5) - n (5 \wedge 7) -n (3 \wedge 7) + n (3 \wedge 5 \wedge 7) = floor(500 1 3 )+ floor (50015) + floor(50017) - floor (500115) - floor (500/35) -floor(500121) + floor (500/105) = 166 + 100 + 71 -33-14-23+4 = 271 This solution is contributed by Sumouli Chaudhary.

Alternate solution

Let a = number divisible by 3

b = number divisible by 5

c = number divisible by 7

n(a) = 166

n(b) = 100

n(c) = 71

n(a∩b) = number divisible by 15 = 33

n(b∩c) = number divisible by 35 = 14

n(a∩c) = number divisible by 21 = 23

n(a∩b∩c) = number divisible by 105 = 4

n(a∪b∪c) = n(a) + n(b) + n(c) - n(a∩b) - n(b∩c) - n(a∩c) + n(a∩b∩c) = 166 + 100 + 71 - 33 - 14 - 23 + 4 = 271

This explanation is contributed by

__Deepak Raj__deepuckraj

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