If the greatest integer k for which 3^k is a… | GMAT Sprint

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PS · Problem Solving

705-805

Divisibility/Multiples/Factors

If the greatest integer k for which $3^k$ is a factor of n! is 8, what is the largest possible value of p so that $5^p$ is a factor of n!?

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Official Explanation

Official

If the greatest integer k for which

3^{k}

is a factor of n! is 8, we need to find the largest possible value of p so that

5^{p}

is a factor of n!.

We can solve by examining factorials:

-

3!

has one

3 = 3^1

6!

has two

3 = 3^2

9!

has four

3 = 3^4

12!

has five

3 = 3^5

15!

has six

3 = 3^6

18!

has eight

3 = 3^8

20!

also has eight

3 = 3^8

21!

has nine

3 = 3^9

Given that the greatest integer k for which

3^{k}

is a factor of

n!

is 8,

n!

cannot be

21!

(since that would give k=9). Therefore,

n!

will be

20!

(or a factorial between 18! and 20!). To maximize the value of p for

5^p

, we take the largest possible n! that still satisfies k=8, which is

20!

.…

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What this question tests

What it tests

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

Common trap

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

If the greatest integer k for which 3k3^k3k is a factor of n! is 8, what is the largest possible value of p so that 5p5^p5p is a factor of n!?