Q58 The number of integers between 1 and 500 both inclusive that are divisible by 3 or 5 or 7 is gate computer science 2017

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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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