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
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