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# If x = 30! + 23, which of the following is a factor of x?

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Joined: 24 Sep 2018
Posts: 137
If x = 30! + 23, which of the following is a factor of x?  [#permalink]

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Updated on: 07 Oct 2018, 03:18
2
00:00

Difficulty:

15% (low)

Question Stats:

77% (00:31) correct 23% (01:09) wrong based on 53 sessions

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If x=30!+23, which of the following is a factor of x?

I. 17
II. 23
III. 29

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

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Originally posted by GMATYoda on 06 Oct 2018, 11:13.
Last edited by Bunuel on 07 Oct 2018, 03:18, edited 1 time in total.
Renamed the topic.
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Joined: 07 Dec 2017
Posts: 1140
Re: If x = 30! + 23, which of the following is a factor of x?  [#permalink]

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06 Oct 2018, 13:08
GMATYoda wrote:
If x=30!+23, which of the following is a factor of x?
I. 17
II. 23
III. 29

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

As all we're given are equations, we'll work with them.
This is a Precise approach.

Every once in a while you see a GMAT question which asks about 'a factorial plus a number'. Usually, what you need to do in this case is to take out a common factor.

In this case, 30!+23 = 30*29*28....*23*22*....*3*2*1 + 23 = 23(30*29*.24*23*21*....1 + 1).
This is clearly divisible by 23 so II is true. how about 17 and 29? Well, for similar reasons we know that 30! + 17 is divisible by 17 meaning that x - 6 is divisible by 17. But since a multiple of 17 occurs every 17 integers, this implies that x is not divisible by 17! Similarly, x+6 = 30!+29 is divisible by 29 meaning that x is not divisible by 29.

So only II is true and (B) is our answer.
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Re: If x = 30! + 23, which of the following is a factor of x?   [#permalink] 06 Oct 2018, 13:08
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