If y is the highest power of 'x' that can divide 101! without leaving a remainder, then for which among the following values of x will y be the highest?

Correct answer marked below

111

462

210

Correct

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Explanation

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Official Explanation

Official

We need to scan the answer choices and check their factors. We want small factors since big factors would give low y.

A: 111 =

3 \times 37

. 37 is a large prime, probably not A.

B: 462 =

6 \times 77 = 2 \times 3 \times 7 \times 11

. y is determined by the number of 11's in 101!, since 2, 3, and 7 appear way more often than 11's.

C: 74 =

2 \times 37

. 37 is too big. This choice eliminates A and C, because A and C give the same y.…

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What this question tests

Your understanding of divisibility rules, least common multiples, and greatest common factors.

Forgetting that 1 is not prime and that every integer divides 0.