GATE CS 2026 Set 1 Q.22: Number Representations and Computer Arithmetic

Consider the 8-bit signed integers $X$ , $Y$ and $Z$ represented using the sign-magnitude form. The binary representations of $X$ and $Y$ are as follows:

X: 10110100 \qquad Y: 01001100

Which of the following operations to compute $Z$ result(s) in an arithmetic overflow?

$Z = X + Y$

$Z = X - Y$

$Z = -X + Y$

$Z = -X - Y$

Official answer and solution

Official answer: B and C
From: Official GATE 2026 answer key

Solution

Convert $X$ and $Y$ from 8-bit sign-magnitude form to decimal and check which results lie outside the 8-bit sign-magnitude range $[-127, +127]$ .

In 8-bit sign-magnitude representation, the MSB is the sign bit ( $0$ for $+$ , $1$ for $-$ ) and the remaining $7$ bits give the magnitude in $[0, 127]$ , so the representable range is $[-127, +127]$ :

$X = 10110100_2$ : sign $= 1$ ( $-$ ), magnitude $= 0110100_2 = 32 + 16 + 4 = 52 \implies X = -52$ .
$Y = 01001100_2$ : sign $= 0$ ( $+$ ), magnitude $= 1001100_2 = 64 + 8 + 4 = 76 \implies Y = +76$ .

Checking each operation:

(A) $Z = X + Y = -52 + 76 = +24 \in [-127, +127]$ (no overflow).
(B) $Z = X - Y = -52 - 76 = -128 \notin [-127, +127]$ (overflow).
(C) $Z = -X + Y = +52 + 76 = +128 \notin [-127, +127]$ (overflow).
(D) $Z = -X - Y = +52 - 76 = -24 \in [-127, +127]$ (no overflow).

Final answer

Options (B), (C)

Common trap

Using the 2's complement range $[-128, +127]$ instead of the sign-magnitude range $[-127, +127]$ , which would miss $-128$ .

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2026 Computer Science and Information Technology - Session 1 (CS1)
Question: 22 of 65
Section: Computer Science and Information Technology

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