Evaluating statements related to remainders of polynomial expressions
Question
A Question is given followed by two Statements I and II. Consider the Question and the Statements. Question: What are the unique values of x and y, where x, y are distinct natural numbers? Statement-I: x/y is odd. Statement-II: xy = 12
Options
The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone
The Question can be answered by using either Statement alone
The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
The Question cannot be answered even by using both the Statements together
Explanation
Analyze the conditions for data sufficiency independently:
Statement I indicates x/y is odd. An infinite number of distinct natural pairs can yield an odd quotient (e.g., 3/1, 6/2, 5/1). Insufficient.
Statement II states xy = 12. Since x and y are distinct natural numbers, list all possible factorization pairs (x, y): (12, 1), (1, 12), (6, 2), (2, 6), (4, 3), (3, 4). Because multiple unique pairs exist, Statement II alone is insufficient.
Combine both statements: We need pairs from Statement II where x/y evaluates to an odd integer:
The only pair satisfying both rules is x = 6 and y = 2. This isolates a single unique solution. Thus, both statements together are sufficient.
Answer: (c).
Question details
Year
2024
Paper
CSAT
Question
Q64
Section
Numerical Ability
Sub-topic
Remainder Theorem
Type
Statement-based
Difficulty
Hard
Source hint
Number theory polynomial remainders
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