If p and q are positive integers, what is the value of q?

Target question: What is the value of q?

Statement 1: \(q^{p-1} = 1\)
Notice that, if \(p = 1\), then \(p-1 = 0\), which means \(q\) can have infinitely many values.
For example consider these possible cases:
Case a: p = 1 and q = 4. These values satisfy the condition that \(q^{p-1} = 1\). In this case, q=4
Case b: p = 1 and q = 3. These values satisfy the condition that \(q^{p-1} = 1\). In this case, q=3
Since we can’t answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: \(p = 1\)
Important: In my two counter-examples above, I let \(p = 1\)
This means we can reuse the same values to show that statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Important: Notice that I was able to use the same counter-examples to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED.
In other words,
Case a: p = 1 and q = 4. These values satisfy the condition that \(q^{p-1} = 1\). In this case, q=4
Case b: p = 1 and q = 3. These values satisfy the condition that \(q^{p-1} = 1\). In this case, q=3
Since we can’t answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent