If y is a positive integer, is y prime?

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.

Answer: B.

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

Hope it helps.

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.

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

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

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

Answer: B

Cheers,
Brent

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.

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.

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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