Finding the smallest number satisfying multiple remainder conditions
Question
Let X be a two-digit number and Y be another two-digit number formed by interchanging the digits of X. If (X + Y) is the greatest two-digit number, then what is the number of possible values of X?
Options
2
4
6
8
Explanation
Let X = 10a + b, where a and b are single-digit integers (a \ne 0). Interchanging the digits gives Y = 10b + a (b \ne 0 for Y to remain a valid two-digit number) [cite: 1567, 1614]. The sum is: X + Y = (10a + b) + (10b + a) = 11(a + b) .
We are given that (X + Y) is the greatest two-digit number, which is 99[cite: 1568, 1614]. 11(a + b) = 99 \implies a + b = 9.
Identify all valid single-digit pairs (a, b) such that a \ge 1 and b \ge 1: (1, 8), (2, 7), (3, 6), (4, 5), (5, 4), (6, 3), (7, 2), (8, 1). There are exactly 8 valid configurations for X.
Answer: (d).
Question details
Year
2024
Paper
CSAT
Question
Q35
Section
Numerical Ability
Sub-topic
LCM & HCF
Type
Factual single
Difficulty
Medium
Source hint
LCM and remainder application
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