Number Systems • Diophantine & Integral Solutions Practice Drills (64 Qs)• Official Answer Keys

CAT Diophantine & Number Theory Integral Solutions Questions (60+ Questions)

Linear Diophantine Equations ax + by = c, Non-Negative Solutions, and Bounded Integer Constraints

64 Total Questions

MCQ: 43 (+3 / -1)

TITA: 21 (0 Negative Penalty)

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Official Exam Fact Sheet & Direct Answer Snapshot

Fact-Checked Archive

CAT Diophantine & Number Theory Integral Solutions Questions (60+ Questions): This official paper archive contains 64 verified CAT questions curated directly from IIM convenor response sheets. The marking scheme follows standard CAT rules: +3 marks for correct MCQs, -1 mark for incorrect MCQs, and zero negative marking penalty for 21 TITA questions. All solutions provide step-by-step mathematical derivations, official convenor response keys, and full 1:1 TCS iON CBT simulator compatibility.

Curriculum DomainCAT Number Systems

Target Solving Speed2.0 to 2.5 mins / question

Exam Weightage1 Question in QA

Testing PlatformTCS iON CBT Simulator

Audited & Verified by ThinkCAT Academic & Editorial Council (Lead: Balaji Rao, IIM Alumnus)Response Sheet Master Key Mapped

Core Formulas & Shortcut Matrix: Diophantine & Integral Solutions Practice Drills

Verified LaTeX Theorems

Existence of Solution to ax + by = cFormula #1

\gcd(a, b) \mid c

Integer solutions exist if and only if GCD of a and b divides c.

Exam Hall Traps & Speedbreakers to Avoid

•Forgetting that non-negative integer solutions include ZERO (0, 1, 2...), while positive requires \ge 1.

Bite-Sized Practice Sets (8 Modules Available)

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Set 03: Integral Solutions

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Set 04: Integral Solutions

Difficulty: Hard

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Set 08: Integral Solutions

Difficulty: Hard

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Official Exam Questions & Explanations (25 of 64)

Sorted in official convenor sequence

Question 1 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsHard

For which of the following values of

k

does the equation

x^2 - 7x + k = 0

have integral solutions?

Practice More Questions

Official Convenor Answer Key: A

Verified Solution: The sum of the roots of the quadratic equation

x^2 - 7x + k = 0

is 7. Let the roots be

a

and

b

. Then,

a + b = 7

and

ab = k

. For

k

to be an integer,

a

and

b

must be integers. The pairs of integers that add up to 7 are $(1, 6) , (2, 5) , (3, 4) , and their negatives. Checking these, $1 \cdot 6 = 6 , 2 \cdot 5 = 10 , 3 \cdot 4 = 12

. Hence,

k = 6$ is the only valid option.

Question 2 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsHard

How many ordered pairs of integers

(x, y)

satisfy the equation

x^2 - 3xy + 2y^2 = 1

Practice More Questions

Official Convenor Answer Key: B

Verified Solution: Factorize the equation:

x^2 - 7x + k = 0

. The factor pairs of 1are

(1, 1)

and

a

. Solving these, we get

(x, y) = (2, 1), (1, 1), (0, -1), (-1, -1)

. Hence, there are 4solutions.

Question 3 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsHard

For which value of

k

does the equation

2x^2 - 3xy + y^2 - 6x + 5y - k = 0

have integral solutions?

Practice More Questions

Official Convenor Answer Key: C

Verified Solution: Rewrite the equation as

x^2 - 7x + k = 0

. Let

2x - y = a , then

y = 2x - a

. Substituting, we get

a^2 - 7x + 10x - 5a - k = 0 ⇒ a^2 + 3x - 5a - k = 0$. For integral solutions, $k$ must be such that the equation can be satisfied. Checking, for $k = 8 , we can find integral solutions.

Question 4 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsHard

For which value of

k

does the equation

x^2 - 5xy + 6y^2 = k

have integral solutions?

Practice More Questions

Official Convenor Answer Key: C

Verified Solution: Factorize the equation:

x^2 - 7x + k = 0

. For

k

to have integral solutions, it must be a product of two integers that are 1apart. Checking,

k = 9 = 3 \times 3

is a valid option.

Question 5 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsHard

For which value of

k

does the equation

x^2 - 4xy + 4y^2 = k

have integral solutions?

TITA Answer:

Practice More Questions

Official Convenor Answer Key: 4

Verified Solution: Factorize the equation:

x^2 - 7x + k = 0

. For

k

to have integral solutions,

k

must be a perfect square. Checking,

k = 9 = 3 \times 3

is a valid option.

Question 6 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsHard

If $x^2 - 5x + 6 = 0 , find the sum of all integral values of $x$ that satisfy the equation $3x^2 - 17x + 20 = 0$.

TITA Answer:

Practice More Questions

Official Convenor Answer Key: 6.00

Verified Solution: First, solve

x^2 - 7x + k = 0

to get

a

. Then substitute these values into

3x^2 - 17x + 20 = 0

. Only

x = 2

satisfies the second equation. Sum of integral solutions is 2.

Question 7 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsHard

Find the sum of all integral values of

x

that satisfy the equation

x^2 - 8x + 15 = 0

and

2x - 7

is a perfect square.

Practice More Questions

Official Convenor Answer Key: A) 5

Verified Solution: Solve

x^2 - 7x + k = 0

to get

a

. Only

3x^2 - 17x + 20 = 0

makes

2x - 7 = 1^2

. Sum of integral solutions is 3.

Question 8 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsHard

Find the sum of all integral values of

x

that satisfy the equation

x^2 - 6x + 8 = 0

and

x^2 - 10x + 21 = 0

TITA Answer:

Practice More Questions

Official Convenor Answer Key: 8.00

Verified Solution: Solve both equations to get

x^2 - 7x + k = 0

for the first and

x = 3, 7

for the second. Only

x = 3

and

x = 7

satisfy both. Sum of integral solutions is 10.

Question 9 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsHard

Find the sum of all integral values of

x

that satisfy the equation

x^2 - 7x + 10 = 0

and

x^2 - 11x + 28 = 0

Practice More Questions

Official Convenor Answer Key: B) 8

Verified Solution: Solve both equations to get

x^2 - 7x + k = 0

for the first and

x = 3, 7

for the second. Only

x = 3

and

x = 7

satisfy both. Sum of integral solutions is 11.

Question 10 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsHard

How many positive integral solutions does the equation

x + 2y + 3z = 30

have?

Practice More Questions

Official Convenor Answer Key: C

Verified Solution: By setting

x^2 - 7x + k = 0

x + 3z = 30$ with solutions $(x, z) = (0, 10), (3, 9), \ldots, (30, 0)$. For each value of $z , x$ must be a multiple of 3. When $y = 1 , we have $x + 3z = 28 , and so on. The pattern continues, and there are 17 solutions in total.

Question 11 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsHard

How many integral solutions does the equation

x^2 + y^2 = 100

have?

Practice More Questions

Official Convenor Answer Key: B

Verified Solution: The pairs

x^2 - 7x + k = 0

that satisfy the equation are

(10, 0), (0, 10), (-10, 0), (0, -10), (6, 8), (6, -8), (-6, 8), (-6, -8), (8, 6), (8, -6), (-8, 6), (-8, -6)

. There are 8 such pairs.

Question 12 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsHard

How many positive integral solutions does the equation

2x + 3y = 30

have?

Practice More Questions

Official Convenor Answer Key: B

Verified Solution: By setting

x^2 - 7x + k = 0

2x = 30 , and

x = 15

. For each value of

y , x

must be such that

2x

is a multiple of 2 and

30 - 3y

is even. The valid pairs are

k

. There are 6 solutions.

Question 13 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsHard

How many positive integral solutions does the equation

3x + 4y = 36

have?

Practice More Questions

Official Convenor Answer Key: D

Verified Solution: By setting

x^2 - 7x + k = 0

3x = 36 , and

x = 15

. For each value of

y , x

must be such that

2x

is a multiple of 3 and

30 - 3y

is a multiple of 3. The valid pairs are

k

. There are 6 solutions.

Question 14 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsEasy

How many integral solutions does the equation

x + y = 5

have?

Practice More Questions

Official Convenor Answer Key: C

Verified Solution: The solutions are (0,5), (1,4), (2,3), (3,2), (4,1), and (5,0). There are 6 solutions.

Question 15 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsModerate

How many pairs of integers

(x, y)

satisfy

x^2 - y^2 = 16

Practice More Questions

Official Convenor Answer Key: B

Verified Solution: Rewrite as

x^2 - 7x + k = 0

. Possible factor pairs are (1,16), (2,8), (4,4), (-1,-16), (-2,-8), (-4,-4). Each pair gives a solution. There are 4 pairs.

Question 16 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsHard

How many solutions does the equation

x + 2y = 10

have where

x

and

y

are positive integers?

Practice More Questions

Official Convenor Answer Key: A

Verified Solution: Solutions are (2,4), (4,3), (6,2), (8,1). There are 4 solutions.

Question 17 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsEasy

How many pairs of integers

(x, y)

satisfy

x + y = 7

Practice More Questions

Official Convenor Answer Key: B

Verified Solution: Solutions are (0,7), (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), (7,0). There are 7 pairs.

Question 18 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsModerate

How many pairs of integers

(x, y)

satisfy

2x + y = 12

Practice More Questions

Official Convenor Answer Key: B

Verified Solution: Solutions are (0,12), (1,10), (2,8), (3,6), (4,4), (5,2), (6,0). There are 7 pairs.

Question 19 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsHard

How many solutions does the equation

x + 3y = 15

have where

x

and

y

are positive integers?

Practice More Questions

Official Convenor Answer Key: A

Verified Solution: Solutions are (3,4), (6,3), (9,2), (12,1). There are 4 solutions.

Question 20 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsEasy

Find the number of integral solutions to the equation

x + y = 10

TITA Answer:

Practice More Questions

Official Convenor Answer Key: 9

Verified Solution: For integral solutions,

x^2 - 7x + k = 0

can take values from 0 to 10. If

x = 0 , y = 10

; if

x = 1 , y = 9

; and so on. Thus, there are 11 pairs, but we exclude the pair (10,0) as it's the same as (0,10) in terms of distinct solutions. Hence, there are 9 unique integral solutions.

Question 21 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsModerate

How many integral solutions does the equation

2x + 3y = 12

have?

TITA Answer:

Practice More Questions

Official Convenor Answer Key: 4

Verified Solution: Rewrite the equation as

x^2 - 7x + k = 0

. For

a

to be an integer,

b

must be divisible by 3. This happens for

x = 0, 3, 6, 9

. Thus, there are 4 solutions.

Question 22 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsHard

Find the number of integral solutions to the equation

x^2 - 5x + 6 = 0

TITA Answer:

Practice More Questions

Official Convenor Answer Key: 2

Verified Solution: Solve the quadratic equation

x^2 - 7x + k = 0

. Factoring, we get

a

. Hence,

x = 2

k

. There are 2 integral solutions.

Question 23 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsEasy

Find the number of integral solutions to the equation

x - y = 5

TITA Answer:

Practice More Questions

Official Convenor Answer Key: 6

Verified Solution: For integral solutions,

x^2 - 7x + k = 0

can take values from -2 to 3. If

x = 0 , y = 10

; if

x = -1 , y = -6

; and so on. Thus, there are 6 unique integral solutions.

Question 24 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsModerate

How many integral solutions does the equation

x + 2y = 11

have?

TITA Answer:

Practice More Questions

Official Convenor Answer Key: 6

Verified Solution: Rewrite the equation as

x^2 - 7x + k = 0

. For

y

to be an integer,

y , x

must be divisible by 2. This happens for

x = 1, 3, 5, 7, 9, 11

. Thus, there are 6 solutions.

Question 25 of 64

ThinkCAT Practice SetQANumber Systems • Integral SolutionsHard

Find the number of integral solutions to the equation

x^2 - 7x + 10 = 0

TITA Answer:

Practice More Questions

Official Convenor Answer Key: 2

Verified Solution: Solve the quadratic equation

x^2 - 7x + k = 0

. Factoring, we get

a

. Hence,

x = 2

k

. There are 2 integral solutions.

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Frequently Asked Questions about CAT Diophantine & Number Theory Integral Solutions Questions (60+ Questions)

What concepts are covered in the Diophantine & Integral Solutions Practice Drills practice module?

This module covers Integral Solutions with 64 verified problems ranging from core foundation to high-difficulty CAT exam hall level.

What is the recommended solving time for Diophantine & Integral Solutions Practice Drills questions?

The ideal target pace is 2.0 to 2.5 mins / question. Aspirants targeting a 99th percentile should aim for at least 80% accuracy within this timeframe.

Where can I find official past year CAT papers for Diophantine & Integral Solutions Practice Drills?

Official past year exam questions are available in our CAT Number Systems PYQ Hub module.

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CAT Diophantine & Number Theory Integral Solutions Questions (60+ Questions)

Core Formulas & Shortcut Matrix: Diophantine & Integral Solutions Practice Drills

Bite-Sized Practice Sets (8 Modules Available)

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Official Exam Questions & Explanations (25 of 64)

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Frequently Asked Questions about CAT Diophantine & Number Theory Integral Solutions Questions (60+ Questions)

What concepts are covered in the Diophantine & Integral Solutions Practice Drills practice module?

What is the recommended solving time for Diophantine & Integral Solutions Practice Drills questions?

Where can I find official past year CAT papers for Diophantine & Integral Solutions Practice Drills?

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