If y is a positive integer, is y prime?

Question

(1) y>4!
 
(2) 11!-12<y<11!-2

EvaJager · Accepted Answer

(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.

Answer B.

Bunuel · Answer

Similar question to practice: Does the integer k have a factor p such that 14! --> 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. Answer: B. Discussed here: does-the-integer-k-have-a-factor-p-such-that-1-p-k-126735.html Hope it helps.

Bunuel · Answer

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 13! --> 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. Answer: B. Discussed here: if-x-is-an-integer-does-x-have-a-factor-n-such-that-100670.html Hope it helps.

ss58146 · Answer

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.

EvaJager · Answer

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.

ss58146 · Answer

it is not a prime, as the number in the parentheses is greater than 1. I get it now .. Thank you..

riskietech · Answer

If y is a positive integer, is y prime?

1) y > 4!
2) 11! – 12 < y < 11! – 2

Bunuel · Answer

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

BrentGMATPrepNow · Answer

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 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 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 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 Answer: B Cheers, Brent

GMATGuruNY · Answer

Statement 1:
Clearly, many values greater than 4! will be prime, while most will not be prime.
Thus, the answer to the question stem can be YES or NO.
INSUFFICIENT.

Statement 2:
Since the GMAT cannot expect us to prove that any of the integers within the given range ARE prime, all of the integers within the given range must NOT be prime.
Thus, the answer to the question stem is NO.
SUFFICIENT.

B

Basshead · Answer

If y is a positive integer, is y prime?

(1) y>4!

Clearly insufficient.

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

11! - 11 = not prime
y = a multiple of 11

11! - 4 = not prime
y = multiple of 4

SUFFICIENT.

Basshead · Answer

That's an incredible approach.

Is it safe to say that if a question asks for us to determine if a very large number is prime, we can assume the answer is no?

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If y is a positive integer, is y prime?

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