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GATE CSE 2026 Session 2 Q26: Properties of the relation 'xy is a perfect square'

Question

Let RR be a binary relation on the set {1,2,…,10}\{1, 2, \ldots, 10\}, where (x,y)∈R(x, y) \in R if the product of xx and yy is square of an integer. Which of the following properties is/are satisfied by RR?

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Answer: A, B, C (official GATE 2026 final answer key)

Concept used: xyxy is a perfect square exactly when xx and yy have the same square-free part.

First step: Write each number as (square) × (square-free part) and see that RR groups numbers by square-free part; test antisymmetry with (1,4)(1, 4).

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Source: official GATE 2026 Session 2 CS question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key

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