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Director  Joined: 03 Sep 2006
Posts: 639
If y is a positive integer, is y prime?  [#permalink]

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Question Stats: 53% (01:31) correct 47% (01:55) wrong based on 193 sessions

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If y is a positive integer, is y prime?

(1) y>4!

(2) 11!-12<y<11!-2

Originally posted by LM on 24 Oct 2012, 05:02.
Last edited by Bunuel on 24 Oct 2012, 05:45, edited 1 time in total.
Edited the question.
Director  Joined: 22 Mar 2011
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Re: If y is a positive integer, is y prime?  [#permalink]

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LM wrote:
If y is a positive integer, is y prime?
1.$$y>4!$$

2.$$11!-12<y<11!-2$$

(1) obviously not sufficient. There are many primes greater than 4! as well as non-primes.

(2) y can be one of the integers 11! - 11, 11! - 10, 11! - 9, ... , 11! - 2, 11! - 3.
It is easy to see that all the numbers on the above list are certainly not primes. 11! = 2x3x4x5x6x7x8x9x10x11, and between 11! and the term subtracted, there is in each case a common factor. So, the first number on the list is divisible by 11, the second by 10,..., the last number is divisible by 3.
Sufficient.

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Re: If y is a positive integer, is y prime?  [#permalink]

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LM wrote:
If y is a positive integer, is y prime?

(1) y>4!

(2) 11!-12<y<11!-2

Similar question to practice:

Does the integer k have a factor p such that 1<p<k?

Question basically asks whether $$k$$ is a prime number. If it is, then it won't have a factor $$p$$ such that $$1<p<k$$ (definition of a prime number).

(1) $$k>4!$$ --> $$k$$ is more than some number ($$4!=24$$). $$k$$ may or may not be a prime. Not sufficient.

(2) $$13!+2\leq{k}\leq{13!+13}$$ --> $$k$$ can not be a prime. For instance if $$k=13!+8=8*(2*4*5*6*7*9*10*11*12*13+1)$$, then $$k$$ is a multiple of 8, so not a prime. Same for all other numbers in this range. So, $$k=13!+x$$, where $$2\leq{x}\leq{13}$$ will definitely be a multiple of $$x$$ (as we would be able to factor out $$x$$ out of $$13!+x$$, the same way as we did for 8). Sufficient.

Discussed here: does-the-integer-k-have-a-factor-p-such-that-1-p-k-126735.html

Hope it helps.
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Re: If y is a positive integer, is y prime?  [#permalink]

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LM wrote:
If y is a positive integer, is y prime?

(1) y>4!

(2) 11!-12<y<11!-2

Another similar question:

If x is an integer, does x have a factor n such that 1 < n < x?

Question basically asks: is $$x$$ a prime number? If it is, then it won't have a factor $$n$$ such that $$1<n<x$$ (definition of a prime number).

(1) $$x>3!$$ --> $$x$$ is more than some number (3!). $$x$$ may or may not be a prime. Not sufficient.

(2) $$15!+2\leq{x}\leq{15!+15}$$ --> $$x$$ can not be a prime. For instance if $$x=15!+8=8*(2*3*4*5*6*7*9*10*11*12*13*14*15+1)$$, then $$x$$ is a multiple of 8, so not a prime. Same for all other numbers in this range: $$x=15!+k$$, where $$2\leq{k}\leq{15}$$ will definitely be a multiple of $$k$$ (as weould be able to factor out $$k$$ out of $$15!+k$$). Sufficient.

Discussed here: if-x-is-an-integer-does-x-have-a-factor-n-such-that-100670.html

Hope it helps.
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Re: If y is a positive integer, is y prime?  [#permalink]

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EvaJager wrote:
LM wrote:
If y is a positive integer, is y prime?
1.$$y>4!$$

2.$$11!-12<y<11!-2$$

(1) obviously not sufficient. There are many primes greater than 4! as well as non-primes.

(2) y can be one of the integers 11! - 11, 11! - 10, 11! - 9, ... , 11! - 2, 11! - 3.
It is easy to see that all the numbers on the above list are certainly not primes. 11! = 2x3x4x5x6x7x8x9x10x11, and between 11! and the term subtracted, there is in each case a common factor. So, the first number on the list is divisible by 11, the second by 10,..., the last number is divisible by 3.
Sufficient.

Hi Eva, can you help to explain me why B is the answer? you mentioned that, "y can be one of the integers 11! - 11, 11! - 10, 11! - 9, ... , 11! - 2, 11! - 3." and "the first number on the list is divisible by 11, the second by 10,..., the last number is divisible by 3", can you help to discuss it in more detail? thanks.
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Director  Joined: 22 Mar 2011
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Re: If y is a positive integer, is y prime?  [#permalink]

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ss58146 wrote:
EvaJager wrote:
LM wrote:
If y is a positive integer, is y prime?
1.$$y>4!$$

2.$$11!-12<y<11!-2$$

(1) obviously not sufficient. There are many primes greater than 4! as well as non-primes.

(2) y can be one of the integers 11! - 11, 11! - 10, 11! - 9, ... , 11! - 2, 11! - 3.
It is easy to see that all the numbers on the above list are certainly not primes. 11! = 2x3x4x5x6x7x8x9x10x11, and between 11! and the term subtracted, there is in each case a common factor. So, the first number on the list is divisible by 11, the second by 10,..., the last number is divisible by 3.
Sufficient.

Hi Eva, can you help to explain me why B is the answer? you mentioned that, "y can be one of the integers 11! - 11, 11! - 10, 11! - 9, ... , 11! - 2, 11! - 3." and "the first number on the list is divisible by 11, the second by 10,..., the last number is divisible by 3", can you help to discuss it in more detail? thanks.

The integers between x - 12 and x - 2 are x - 11, x - 10, ..., x - 3. In our case x =11!.

Take common factor between 11! and the term that is subtracted from it:
For example, 11! - 11 = 2x3x4x5x6x7x8x9x10x11 - 11 = 11(2x3x4x5x6x7x8x9x10 - 1) is divisible by 11, so it is not a prime, as the number in the parentheses is greater than 1.
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Re: If y is a positive integer, is y prime?  [#permalink]

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it is not a prime, as the number in the parentheses is greater than 1. I get it now .. Thank you.. _________________
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If y is a positive integer, is y prime?  [#permalink]

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If y is a positive integer, is y prime?

1) y > 4!
2) 11! – 12 < y < 11! – 2
Math Expert V
Joined: 02 Sep 2009
Posts: 58449
Re: If y is a positive integer, is y prime?  [#permalink]

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riskietech wrote:
If y is a positive integer, is y prime?

1) y > 4!
2) 11! – 12 < y < 11! – 2

Merging similar topics. Please refer to the discussion above.

Hope it helps.

Similar questions to practice:
does-the-integer-k-have-a-factor-p-such-that-1-p-k-126735.html
if-z-is-an-integer-is-z-prime-128732.html
if-x-is-an-integer-does-x-have-a-factor-n-such-that-100670.html
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Re: If y is a positive integer, is y prime?  [#permalink]

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Top Contributor
LM wrote:
If y is a positive integer, is y prime?

(1) y > 4!

(2) 11! - 12 < y < 11! - 2

Target question: is y prime?

Statement 1: y > 4!
In other words, y > 24
This does not help us determine whether or not y is prime. Consider these two conflicting cases:
Case a: y = 29, in which case y IS prime
Case b: y = 25, in which case y is NOT prime
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: 11! – 12 < y < 11! – 2
Let's examine a few possible values for y.

y = 11! – 11
y = (11)(10)(9)....(5)(4)(3)(2)(1) - 11
y = 11[(10)(9)....(5)(4)(3)(1) - 1]
Since y is a multiple of 11, y is NOT prime

y = 11! – 10
y = (11)(10)(9)....(5)(4)(3)(2)(1) - 10
y = 10[(11)(9)....(5)(4)(3)(1) - 1]
Since y is a multiple of 10, y is NOT prime

y = 11! – 9
y = (11)(10)(9)....(5)(4)(3)(2)(1) - 9
y = 9[(11)(10)....(5)(4)(3)(1) - 1]
Since y is a multiple of 9, y is NOT prime

As you can see, this pattern can be repeated all the way up to y = 11! - 1. In EVERY case, y is NOT prime
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Cheers,
Brent
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Re: If y is a positive integer, is y prime?  [#permalink]

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_________________ Re: If y is a positive integer, is y prime?   [#permalink] 06 Jun 2019, 06:29
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