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# ‘x’ is a positive integer. Is (x-1)! + 1 divisible by x?

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CrackVerbal Quant Expert
Joined: 12 Apr 2019
Posts: 127
‘x’ is a positive integer. Is (x-1)! + 1 divisible by x?  [#permalink]

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17 May 2019, 06:29
00:00

Difficulty:

95% (hard)

Question Stats:

23% (02:34) correct 77% (02:12) wrong based on 30 sessions

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‘x’ is a positive integer. Is (x-1)! + 1 divisible by x?

I. x is an odd number greater than 2 but less than 10

II. x is a prime number

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Joined: 02 Aug 2009
Posts: 7765
Re: ‘x’ is a positive integer. Is (x-1)! + 1 divisible by x?  [#permalink]

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19 May 2019, 03:57
x is a positive integer. Is (x-1)! + 1 divisible by x?
If x is a prime number, Ans is always yes. When x is not a prime, answer will be NO in most of the cases

I. x is an odd number greater than 2 but less than 10
When x is 3,5 or 7 answer is yes and when x is 9, answer is No.

II. x is a prime number
Sufficient

B
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Re: ‘x’ is a positive integer. Is (x-1)! + 1 divisible by x?  [#permalink]

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19 May 2019, 04:12
X=Prime then
X-1)!/X remainder will be always be X-1
And as per problem (X-1)! +1 will always be divisible by X Oly for prime

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CrackVerbal Quant Expert
Joined: 12 Apr 2019
Posts: 127
Re: ‘x’ is a positive integer. Is (x-1)! + 1 divisible by x?  [#permalink]

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21 May 2019, 06:34
2
This is a moderately difficult problem that tests your knowledge on a specific remainder concept. This concept is,

"If ‘x’ is a prime number, (x-1)! will leave a remainder of (x-1) when divided by x. "

Put in simple words, this means that (x-1)! + 1 will be a multiple of x only if x is prime. Hence, in this question, we are essentially trying to find out if ‘x’ is prime.

From statement I alone, x can be 3 or 5 or 7 or 9. We cannot answer the main question with certainty, because 3,5, and 7 are prime while 9 is not. So, statement I alone is insufficient.
Answer options A and D can be ruled out, possible answer options are B, C or E.

From statement II alone, there cannot be any doubt at all. The statement says that ‘x’ is prime and this is what we are trying to find out.
In other words, we can say for sure that, since x is a prime numbers, (x-1)! + 1 will be divisible by x. Statement II alone is hence sufficient.

Therefore, the correct answer option is B.

If you were not familiar with the concept mentioned above, the alternative approach is to try values and see if a pattern develops. With statement I, you only have to try 4 values.
But, when it comes to statement II, it’s natural to worry about how many values you need to try, because there are infinitely many prime numbers. However, the fact about GMAT problems is, if a pattern is not there, it will become apparent to you in the first few instances itself; it’s not that the pattern breaker will reveal itself after 20 values. So, if you see a pattern developing after 4 or 5 values, go ahead and mark the answer.

Of course, the more foolproof of the two methods is the one using concepts. The best method, perhaps, is a combination of the both. Don’t you think so!

Hope this helps!
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Re: ‘x’ is a positive integer. Is (x-1)! + 1 divisible by x?   [#permalink] 21 May 2019, 06:34
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