Bunuel wrote:
If \(n\) is a positive integer and \(p\) is a prime number, is \(p\) a factor of \(n!\)?
(1) \(p\) is a factor of \((n+2)!-n!\)
(2) \(p\) is a factor of \(\frac{(n+2)!}{n!}\)
Target question: Is p a factor of n! Statement 1: \(p\) is a factor of \((n+2)!-n!\) \((n+2)!=(n+2)(n+1)(n)(n-1)(n-2)...(3)(2)(1)\)
And \(n!=(n)(n-1)(n-2)...(3)(2)(1)\)
So, we can factor the expression to get: \(p\) is a factor of \(n![(n+2)(n+1)-1]\)
So, it COULD be the case that p is a factor of n!, in which case, the answer to the target question is
YES, p IS a factor of n!Or, it COULD be the case that p is a factor of [(n+2)(n+1)-1], in which case, the answer to the target question is
NO, p is NOT a factor of n!ASIDE: If we let n = 2, and p = 2, then we can see that p IS a factor of n!
If we n = 2, and p = 11, then we can see that p is NOT a factor of n!, but it IS a factor of [(n+2)(n+1)-1]
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: \(p\) is a factor of \(\frac{(n+2)!}{n!}\)Simplify the expression to get: \(p\) is a factor of \((n+2)(n+1)\)
Let's test some values that satisfy statement 2:
Case a: n = 2 and p = 2. Here, (n+2)(n+1) = (2+2)(2+1)=12, so 2 (aka p) is a factor of (n+2)(n+1). In this case, the answer to the target question is
YES, p IS a factor of n!Case b: n = 2 and p = 3. Here, (n+2)(n+1) = (2+2)(2+1)=12, so 3 (aka p) is a factor of (n+2)(n+1). In this case, the answer to the target question is
NO, p is NOT a factor of n!Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined Statement 2 tells us that \(p\) is a factor of \((n+2)(n+1)\)
Notice that, if \(p\) is a factor of \((n+2)(n+1)\), then \(p\) is NOT a factor of \((n+2)(n+1)-1\)
Statement 1 tells us that \(p\) is a factor of \(n![(n+2)(n+1)-1]\)
Since we know that \(p\) is NOT a factor of \((n+2)(n+1)-1\), it MUST be true that \(p\) is a factor of \(n!\)
The answer to the target question is
YES, p IS a factor of n!Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer: C
Cheers,
Brent
_________________
Brent Hanneson – Creator of gmatprepnow.com
I’ve spent the last 20 years helping students overcome their difficulties with GMAT math, and the biggest thing I’ve learned is…
Many students fail to maximize their quant score NOT because they lack the skills to solve certain questions but because they don’t understand what the GMAT is truly testing -
Learn more