How many positive integers less than 60 are relatively prime to 60?
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Answer & Explanation
Correct answer
B
Two integers are relatively prime when their greatest common divisor is 1. Factor 60 = 2² × 3 × 5, so an integer is relatively prime to 60 exactly when it is divisible by none of the primes 2, 3, 5.
The count of such integers from 1 to 60 is 60 × (1 − 1/2)(1 − 1/3)(1 − 1/5) = 60 × (1/2)(2/3)(4/5) = 60 × 8/30 = 16.
As a cross-check, count the integers from 1 to 60 that are divisible by 2, 3, or 5 using inclusion-exclusion: multiples of 2, 3, 5 number 30, 20, 12; subtract the multiples of 6, 10, 15 (10, 6, 4); add back the multiples of 30 (2). That gives 30 + 20 + 12 − 10 − 6 − 4 + 2 = 44, leaving 60 − 44 = 16 integers in 1 to 60 that share no factor with 60.
Since 60 itself is not relatively prime to 60, removing it changes nothing, so exactly 16 positive integers less than 60 are relatively prime to 60. Dropping the factor for 5 gives 60 × (1/2)(2/3) = 20, the trap behind choice C.