CSAT Solved Papers/ 2022/Q66

2022 CSAT — Q66

Quant Number theory 2.5 marks Medium

Let $p$ be a two-digit number and $q$ be the number consisting of same digits written in reverse order. If $p \times q = 2430$ , then what is the difference between $p$ and $q$ ?

A 45
B 27
C 18
D 9 Answer

Worked rationale

We need a two-digit $p$ and its reverse $q$ with $p \times q = 2430$ .

Factor $2430 = 2 \cdot 3^5 \cdot 5$ . Scan two-digit factor pairs of $2430$ that are digit-reverses of each other. Test $54 \times 45$ :

54 \times 45 = 2430. \checkmark

and $45$ is the reverse of $54$ . So $p = 54,\ q = 45$ (or vice versa), and

|p - q| = |54 - 45| = 9.

Answer: (d) 9.

Why the other options miss

A

answered the sub-step, not the question: returns the smaller number $q = 45$ instead of the difference.
B

an arithmetic slip: pairs a wrong reverse (e.g. $81$ and $18$ , whose product is $1458$ , not $2430$ ) and reports its difference.
C

an arithmetic slip: uses $63$ and $45$ -type mismatched pair; the difference is wrong because the pair isn’t a true reverse with product $2430$ .

Specialist insight

The difference of a two-digit number and its reverse is always $9\,|a - b|$ where $a, b$ are the digits, so the answer must be a multiple of $9$ . Factor $2430$ and look for the reverse pair: $54 \times 45 = 2430$ with digits $5, 4$ gives $9\,|5-4| = 9$ . Knowing “reverse-difference is a multiple of $9$ ” lets you sanity-check instantly.

The trap, in one line

$54 \times 45 = 2430$ (reverses); $|54 - 45| = 9 \Rightarrow$ (d).

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