CSAT Solved Papers/ 2022/Q67

2022 CSAT — Q67

Quant Statement validity 2.5 marks Medium

Consider the following statements in respect of two natural numbers $p$ and $q$ such that $p$ is a prime number and $q$ is a composite number:

$p \times q$ can be an odd number.
$q / p$ can be a prime number.
$p + q$ can be a prime number.

Which of the above statements are correct?

A 1 and 2 only
B 2 and 3 only
C 1 and 3 only
D 1, 2 and 3 Answer

Worked rationale

Each statement says “can be” — so one witnessing example proves it.

Statement 1 ( $p \times q$ odd): take $p = 3$ (prime), $q = 9$ (composite). $3 \times 9 = 27$ is odd. ✓

Statement 2 ( $q/p$ prime): take $p = 2$ , $q = 6$ (composite). $6 / 2 = 3$ , a prime. ✓

Statement 3 ( $p + q$ prime): take $p = 2$ , $q = 9$ (composite). $2 + 9 = 11$ , a prime. ✓

All three are achievable.

Answer: (d) 1, 2 and 3.

Why the other options miss

A

missed a case: assumes $p + q$ must be even/composite, missing $2 + 9 = 11$ .
B

missed a case: thinks a prime $\times$ composite must be even, forgetting both can be odd ( $3 \times 9$ ).
C

missed a case: doubts $q/p$ can be prime, overlooking $q = p \times (\text{prime})$ like $6/2 = 3$ .

Specialist insight

“Can be” items are existence claims — never argue impossibility from a single failed try; search for one success. The three witnesses ( $3{\times}9 = 27$ odd; $6/2 = 3$ prime; $2{+}9 = 11$ prime) each exploit a different freedom: odd prime $\times$ odd composite, composite $=$ prime $\times$ prime, and the even prime $2$ summed with an odd composite to land on a prime.

The trap, in one line

Witnesses $3{\times}9{=}27$ , $6/2{=}3$ , $2{+}9{=}11$ make all three possible $\Rightarrow$ (d).

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