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GATE 2026 AR — Planning – Question 9

General Aptitude · Quantitative Aptitude: Data Interpretation, Arithmetic and Number Computation · 2 marks · Multiple choice

$P, Q, R, S, X$, and $Y$ are distinct single-digit whole numbers taking values from 0 to 9.

$PQ$ is a two-digit number with $Q$ being in the units place and $P$ in the tens place. Similarly, $RS$ is a two-digit number.

It is known that $PQ$ and $RS$ are consecutive numbers and $(PQ)^2 + (RS)^2 = XYP$, with $XYP$ being a three-digit number.

The value of $Y$ is __________

  1. 4
  2. 5
  3. 6
  4. 7

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Show answer and explanation

Correct answer: (C) 6

Explanation

The sum is a three-digit number. Because 22²+23²=1013 is too large, the greater of the consecutive numbers is at most 22. Thus the first number's tens digit P is either 1 or 2.

Consecutive numbers have opposite parity, so their square sum is odd. Its last digit is P, making P odd; hence P=1.

The tens digits P and R must be distinct. Two consecutive two-digit numbers with different tens digits cross a boundary, so the only pair in this range is 19 and 20. Since P=1 belongs to the first number, PQ=19 and RS=20.

Then 19²+20²=361+400=761. The six digits (P, Q, R, S, X, Y)=(1, 9, 2, 0, 7, 6) are all distinct. Therefore **Y=6, C**. This proves uniqueness rather than merely finding a pair that works.

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