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Number Systems, Digit Problems and Divisibility PYQs for GATE DA

Solve 2+ Number Systems, Digit Problems and Divisibility previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

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Question 1

2026 PYQ

Level 3: Exam Standard

P, Q, R, S, X, and Y are distinct single-digit whole numbers taking values from 0 to 9. P Q is a two-digit number with Q being in the units place and P in the tens place. Similarly, R S is a two-digit number. It is known that P Q and R S are consecutive numbers and (P Q)^{2} + (R S)^{2} = X Y P, with X Y P being a three-digit number. The value of Y is __________

Question 2

2026 PYQ

Level 3: Exam Standard

The product of the digits of a three-digit number is 70. The sum of the digits of this three-digit number is _____

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Number Systems, Digit Problems and Divisibility PYQs for GATE DA

Solve 2+ Number Systems, Digit Problems and Divisibility previous year questions for GATE DA with answers and detailed solutions. Free sample questions below.

Fundamental Rules of Number Formation

When forming an $n$ -digit number, the most significant digit (the leading digit) can never be $0$ .

Basic Counting Rules

1. Leading Zero Constraint: For an

n

-digit number, the first position has restricted choices. If forming a number from a set containing

0

, the first position cannot be

0

2. Repetition Allowed: If digits can be repeated, each subsequent position has the full set of available choices.

3. Repetition Not Allowed: If digits cannot be repeated, each subsequent position has one fewer choice than the previous.

Example

Form a 3-digit number from the set ${0, 1, 2, 3, 4}$ without repetition.

Hundreds place: $4$ choices (cannot be $0$ ).
Tens place: $4$ choices (includes $0$ , but one non-zero digit is used).
Units place: $3$ choices.

Total numbers = $4 \times 4 \times 3 = 48$ .

Applying Divisibility Rules to Formed Numbers

Applying Divisibility Rules

Fix the digits that satisfy the divisibility rule first, then fill the remaining positions.

Div by 2
Unit:

0, 2, 4, 6, 8

Div by 4
Last 2 digits

Div by 5
Unit:

0

5

Worked Example

How many 4-digit numbers from ${1, 2, 3, 4, 5}$ without repetition are divisible by $5$ ?

Step 1 Fix unit digit:

5

(

1

choice).

Step 2 Thousands place:

4

choices.

Step 3 Hundreds place:

3

choices.

Step 4 Tens place:

2

choices.

Total = $1 \times 4 \times 3 \times 2 = 24$

Number Systems, Digit Problems and Divisibility: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Aptitude · 2026 MCQ

P, Q, R, S, X,

and

Y

are distinct single-digit whole numbers taking values from 0 to 9.

P Q

is a two-digit number with

Q

being in the units place and

P

in the tens place. Similarly,

R S

is a two-digit number.
It is known that

P Q

and

R S

are consecutive numbers and

(P Q)^{2} + (R S)^{2} = X Y P,

with

X Y P

being a three-digit number.
The value of

Y

is __________

A.
4
B.
5
C.
6
D.
7

Correct Answer:

Step-by-Step Solution

Insight: Let $P Q = n$ and $R S = n + 1$ (consecutive). Then $n^{2} + (n + 1)^{2} = 2 n^{2} + 2 n + 1 = X Y P$ . The units digit of this expression must equal $P$ , and $P$ is the tens digit of $n$ . This double role of $P$ is the key constraint.

Exam route:

Step 1: $2 n^{2} + 2 n + 1 \equiv P (mod 10)$ , where $n = 10 P + Q$ .

Step 2: Try $P = 1$ (since $n \leq 21$ for $X Y P$ to be 3-digit). $2 n^{2} + 2 n \equiv 0 (mod 10) ⟹ n (n + 1) \equiv 0 (mod 5) ⟹ n \equiv 0$ or $4 (mod 5)$ .

Step 3: $n \in {10, 14, 15, 19}$ . Check each:

$n = 10$ : $221 ⟹ X = 2, Y = 2$ (not distinct).
$n = 14$ : $421 ⟹ X = 4, Y = 2, P = 1, Q = 4$ . But $X = Q = 4$ (not distinct).
$n = 15$ : $481 ⟹ X = 4, Y = 8, P = 1, Q = 5$ . $R S = 16 ⟹ R = 1 = P$ (not distinct).
$n = 19$ : $761 ⟹ X = 7, Y = 6, P = 1, Q = 9$ . $R S = 20 ⟹ R = 2, S = 0$ . Digits ${1, 9, 2, 0, 7, 6}$ all distinct. ✓

Step 4: $Y = 6$ .

Learning route:

The constraint that $P$ is both the tens digit of $P Q$ and the units digit of $X Y P$ is the pivot. We express $X Y P = 100 X + 10 Y + P$ and $P Q = 10 P + Q$ . The equation $n^{2} + (n + 1)^{2} = 100 X + 10 Y + P$ forces $2 n^{2} + 2 n + 1 \equiv P (mod 10)$ . Since $n = 10 P + Q$ , we test small $P$ (as $n \leq 21$ for 3-digit result). Only $n = 19$ yields all distinct digits.

Trap: Ignoring that $P$ must be the tens digit of $P Q$ leads to $n = 20 ⟹ 841 ⟹ Y = 4$ , but $P = 2 \neq = 1$ .

Verification: $1 9^{2} + 2 0^{2} = 361 + 400 = 761$ . $X = 7, Y = 6, P = 1$ . $P Q = 19, R S = 20$ . All digits ${1, 9, 2, 0, 7, 6}$ distinct. ✓

Question 2 · Quantitative Aptitude · 2026 MCQ

The product of the digits of a three-digit number is 70. The sum of the digits of this three-digit number is _____

A.
12
B.
14
C.
16
D.
18

Correct Answer:

Step-by-Step Solution

Insight: The product of three single digits is 70, so factorise 70 into three single-digit factors first; the sum follows immediately.

Exam route: $70 = 2 \times 5 \times 7$ . These are the only three single-digit factors (any other factorisation like $1 \times 7 \times 10$ uses a non-digit). Sum $= 2 + 5 + 7 = 14$ .

Learning route:

Step 1: Prime factorise $70 = 2 \times 5 \times 7$ .

Step 2: We need three single digits $a, b, c \in {1, 2, \dots, 9}$ such that $a \times b \times c = 70$ .

Step 3: Since $70 = 2 \times 5 \times 7$ , and all three are single digits, the only valid triple is ${2, 5, 7}$ . Any attempt to introduce a 1 (e.g. $1 \times 5 \times 14$ ) forces a factor $\geq 10$ , which is not a digit.

Step 4: Sum $= 2 + 5 + 7 = 14$ .

Trap: Using $1 \times 7 \times 10$ gives sum 18, but 10 is not a single digit.

Verification: $2 \times 5 \times 7 = 70$ ✓, and $2 + 5 + 7 = 14$ ✓.

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Unlock the whole course

Number Systems, Digit Problems and Divisibility PYQs for GATE DA

Fundamental Rules of Number Formation

Fundamental Rules of Number Formation

Basic Counting Rules

Example

Applying Divisibility Rules to Formed Numbers

Applying Divisibility Rules

Worked Example

Number Systems, Digit Problems and Divisibility: Solved Questions with Step-by-Step Explanations (2 Problems)

Step-by-Step Solution

Step-by-Step Solution

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