Consider the following in respect of prime number p and composite number c: 1. $\dfrac{p + c}{p - c}$ can be…
Consider the following in respect of prime number p and composite number c:
1. $\dfrac{p + c}{p - c}$ can be even.
2. 2p + c can be odd.
3. pc can be odd.
Which of the statements given above are correct?
1 and 2 only
2 and 3 only
1 and 3 only
1, 2 and 3
Solution
Statement 1: Take $p = 3$, $c = 9$: $(p+c)/(p-c) = 12/(-6) = -2$, which is even. So statement 1 can be true. Statement 2: $2p$ is always even; if $c$ is odd (e.g. $c = 9$), then $2p + c$ is odd. So statement 2 can be true. Statement 3: $pc$ is odd when both $p$ and $c$ are odd (e.g. $p = 3$, $c = 9$, $pc = 27$). So statement 3 can be true. All three statements are correct.