Bunuel wrote:
If \(p\) is a positive integer, is \(p\) a prime number?(1) \(p\) and \(p+1\) have the same number of factors.
(2) \(p-1\) is a factor of \(p\).
Kudos for a correct solution.
Target question: Is p a prime number? Given: p is a positive integer Statement 1: p and p+1 have the same number of factors. Let's TEST some values.
There are several values of p that satisfy statement 1. Here are two:
Case a: p = 2. This means that p+1 = 3. Notice that 2 and 3 both have the same number of factors (2 factors each). In this case,
p IS primeCase b: p = 14. This means that p+1 = 15. Notice that 14 and 15 both have the same number of factors (4 factors each). In this case,
p is NOT primeSince we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: p−1 is a factor of p Let's test some cases:
If p = 3, then p-1 = 2. Is 2 a factor of 3? No.
If p = 4, then p-1 = 3. Is 3 a factor of 4? No.
If p = 5, then p-1 = 4. Is 4 a factor of 5? No.
If p = 6, then p-1 = 5. Is 5 a factor of 6? No.
.
.
.
We can see that, if we keep going, p-1 will NEVER be a factor of p. Yet, statement 2 says that p-1 IS a factor of p.
Let's test the two positive integers that we haven't yet tested: 2 and 1
If p = 2, then p-1 = 1. Is 1 a factor of 2? YES! So,
p COULD equal 2If p = 1, then p-1 = 0. Is 0 a factor of 1? No
So, We can conclude that p MUST equal 2, which means
p IS primeSince we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer: B
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