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. Since 60 = 2² × 3 × 5, an integer is relatively prime to 60 exactly when it shares none of the prime factors 2, 3, or 5.
Count such integers from 1 to 60 with Euler's totient formula: 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 divisible by 2, 3, or 5 using inclusion-exclusion. Multiples of 2, 3, and 5 number 30, 20, and 12. Subtract the double-counted multiples of 6, 10, and 15, which number 10, 6, and 4. Add back the multiples of 30, which number 2. This gives 30 + 20 + 12 − 10 − 6 − 4 + 2 = 44 integers that share a factor with 60, leaving 60 − 44 = 16 that do not.
Since 60 itself shares all three prime factors with 60, it is not relatively prime to 60, so it was already excluded and removing it changes nothing. Exactly 16 positive integers less than 60 are relatively prime to 60.
The correct answer is 16.