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If \(n\) is a prime number and \(n \ne 3,\) which of the following could be the remainder when \(100! + n\) is divided

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by Vincen » Tue Jun 30, 2020 7:23 am

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Answers

A

B

C

D

E

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Difficulty—

If \(n\) is a prime number and \(n \ne 3,\) which of the following could be the remainder when \(100! + n\) is divided by \(3?\)

I. 0
II. 1
III. 2

A. II only
B. III only
C. I and II only
D. II and III only
E. I, II and III

[spoiler]OA=D[/spoiler]

Source: Princeton Review
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Re: If \(n\) is a prime number and \(n \ne 3,\) which of the following could be the remainder when \(100! + n\) is divid

by Jay@ManhattanReview » Wed Jul 01, 2020 1:08 am
Vincen wrote: ↑
Tue Jun 30, 2020 7:23 am
 If \(n\) is a prime number and \(n \ne 3,\) which of the following could be the remainder when \(100! + n\) is divided by \(3?\)

I. 0
II. 1
III. 2

A. II only
B. III only
C. I and II only
D. II and III only
E. I, II and III

[spoiler]OA=D[/spoiler]

Source: Princeton Review
We know that 100! = 1*2*3*...*100. So 100! is divisible by 3; the remainder upon dividing \(100! + n\) by 3 would come from dividing \(n\) by 3.

Knowing that \(n\) is a prime number and \(n \ne 3,\) the remainder cannot be 0. If n = 5, the remainder is 2 and if n = 7, the remainder is 1.

Correct answer: D

Hope this helps!

-Jay
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