Let p be a two-digit number and q be the number consisting of same digits written in reverse order. If p × q = 2430, then what is the difference between p and q?
- (a) 45
- (b) 27
- (c) 18
- (d) 9 ✓ UPSC's answer
Why the answer is (d)
• Let p = 10a + b and q = 10b + a, where a and b are non-zero digits, so that p and q are two-digit reverses of each other.
• The condition is (10a + b)(10b + a) = 2430; factoring 2430 = 2 × 5 × 3⁵, we search for a pair of two-digit factors that are digit-reverses.
• Testing the factor pair 45 and 54: 45 × 54 = 45 × 50 + 45 × 4 = 2250 + 180 = 2430, which satisfies the equation.
• Thus {p, q} = {45, 54}, and the required difference is |54 − 45| = 9.
• The difference between p and q is 9, so the answer is option (d).
Why the other options are wrong
- (a) 45
- 45 is the value of p itself, not the difference between p and q; the difference is 54 − 45 = 9.
- (b) 27
- 27 does not equal |54 − 45|; the actual difference between the two numbers is 9, not 27.
- (c) 18
- 18 does not equal |54 − 45|; the actual difference between the two numbers is 9, not 18.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.