GATE | GATE CS Mock 2018 | Set 2 | Question 59

How many positive integers less than 1000 are coprime with 14?
(A) 571
(B) 142
(C) 429
(D) None of these


Answer: (C)

Explanation: For any number to be coprime with 14, it must not be a multiple of 2 and 7.
Number of integers less than 1000 which are multiples of 2 = 1000/2 = 500
Number of integers less than 1000 which are multiples of 7 = floor(1000/7) = 142
Number of integers less than 1000 which are multiples of 14 = floor(1000/14) = 71
Number of integers less than 1000 which are multiples of 2 or 7 = 500 + 142 – 71 = 571
Number of integers less than 1000 which are not multiples of 2 or 7 = 1000 – 571 = 429

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