Number Theory Word Problems and Diophantine Conditions Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Number Theory Word Problems and Diophantine Conditions notes for CAT: 27 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Number Theory Word Problems and Diophantine Conditions

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CAT QA • Number System

Chapter Roadmap

Step 1 • Selected Topic • PYQ count: 1 • Importance: Moderate

① Diophantine Equations and Natural Number Constraints

You will learn how to solve equations where variables must be natural numbers, distinct, positive, or minimum/maximum satisfying conditions.

Mastery target: turn a scary equation like

3 a c = 8 (a + b)

into divisibility cases.

Step 2 • Context Topic • PYQ count: 1 • Importance: Moderate

② Ratio-Based Divisibility Word Problems

You will later use ratio, multiples, and divisibility conditions in word problems. This is nearby, but not taught in this selected topic.

Boundary note: today we focus only on Diophantine equations with natural-number restrictions.

Why this chapter exists: CAT sometimes hides number theory inside simple-looking equations. The trick is not heavy algebra; it is respecting integer conditions.

Diophantine Equations and Natural Number Constraints

Topic Hero

Diophantine Equations and Natural Number Constraints

The equation is only half the story. The real CAT move is: which integer values are actually allowed?

🎯

What you master

Convert equations into divisibility cases and find the smallest valid expression.

⚠️

Main trap

Treating $a, b, c$ like real numbers and forgetting natural-number restrictions.

CAT signal: words like natural numbers, distinct, integer, least possible, smallest value, number of solutions.

What Makes an Equation Diophantine?

The Meaning

A Diophantine equation is an equation where variables must take integer-type values.

Normal algebra

Find any value satisfying the equation.

Example: $2 x = 5 \Rightarrow x = 2.5$ allowed.

Diophantine thinking

Find only integer or natural-number values.

Example: $2 x = 5$ has no integer solution.

In CAT, this topic usually appears as a short equation plus conditions like:
natural numbers, distinct, positive, smallest possible value.

Natural Numbers Are Not Just Decoration

Natural Number Constraint

For CAT-style problems, read natural numbers as:

$1, 2, 3, 4, \dots$

Allowed

Positive whole numbers: $1, 2, 3, \dots$

Not allowed

$0$ , negative values, fractions, decimals.

Exam habit: after solving, verify every variable separately: positive? integer? distinct? satisfies original equation?

Number Theory Word Problems and Diophantine Conditions: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Ability MCQ

The daily outputs of two machines A and B are in the ratio $3 : 8$ , so they can be written as $3 k$ and $8 k$ for a natural number $k$ . The total output of A in 4 days is a multiple of 7, and the total output of B in 3 days is a multiple of 5. If $k$ and a natural number $t$ satisfy

$2 k + 3 t = 300,$

what is the maximum possible daily output of machine B?

A.
560
B.
840
C.
1120
D.
1400

Correct Answer:

Step-by-Step Solution

Key idea: this is a chapter-synthesis maximum problem. The ratio gives a hidden multiplier, the divisibility conditions restrict $k$ , and the extra Diophantine equation filters the remaining candidates.

Step 1: Write the daily outputs.

A produces $3 k$ per day and B produces $8 k$ per day.

Step 2: Use A's divisibility condition.

A's 4-day total is $12 k$ , and it is a multiple of 7.

Since $g cd (12, 7) = 1$ , $k$ must be a multiple of 7.

Step 3: Use B's divisibility condition.

B's 3-day total is $24 k$ , and it is a multiple of 5.

Since $g cd (24, 5) = 1$ , $k$ must be a multiple of 5.

Step 4: Combine the ratio-divisibility conditions.

$k$ must be a multiple of both 7 and 5, so $k$ is a multiple of

$lcm (7, 5) = 35.$

Step 5: Use the Diophantine equation.

$2 k + 3 t = 300$

Since $t$ is natural, $300 - 2 k$ must be a positive multiple of 3. Also $2 k < 300$ , so

$k < 150.$

Step 6: List multiples of 35 below 150.

$35, 70, 105, 140$

Now require $300 - 2 k$ to be divisible by 3.

Since $300 \equiv 0 (mod 3)$ , we need

$2 k \equiv 0 (mod 3) .$

Because 2 is invertible modulo 3, this means

$k \equiv 0 (mod 3) .$

Among $35, 70, 105, 140$ , only $105$ is divisible by 3.

Step 7: Confirm $t$ .

$t = \frac{300 - 2 ( 105 )}{3} = \frac{90}{3} = 30,$

which is natural.

Step 8: Compute B's daily output.

$8 k = 8 \cdot 105 = 840.$

Answer: B

Question 2 · Quantitative Ability MCQ

Two weekly collections are in the ratio $2 : 9$ . If $k$ is the common natural-number multiplier, which pair correctly represents the first collection and the second collection?

A.
$2 + k$ and $9 + k$
B.
$2 k$ and $9 k$
C.
$9 k$ and $2 k$
D.
$k /2$ and $k /9$

Correct Answer:

Step-by-Step Solution

Key idea: this is a ratio-to-multiplier representation question, recognisable because the problem gives a ratio and asks for the actual form of the quantities.

Step 1: The ratio $2 : 9$ gives the shape of the two quantities. The first quantity corresponds to $2$ , and the second corresponds to $9$ .

Step 2: A ratio does not mean the actual values are exactly $2$ and $9$ . It means both values are obtained by multiplying the ratio terms by the same common multiplier.

Step 3: Let that common multiplier be $k$ . Then the first collection is $2 k$ and the second collection is $9 k$ .

Step 4: Keep the order of the ratio. Since the first term is $2$ , the first quantity is $2 k$ , not $9 k$ .

Answer: $2 k$ and $9 k$ .

More notes in this unit

chapter

Number Theory Word Problems and Diophantine Conditions Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Number Theory Word Problems and Diophantine Conditions notes for CAT: 27 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practi

A question from this chapter

Question 1

The daily outputs of two machines A and B are in the ratio 3 : 8, so they can be written as 3 k and 8 k for a natural number k . The total output of A in 4 days is a multiple of 7, and the total output of B in 3 days is a multiple of 5. If k and a natural number t satisfy 2 k + 3 t = 300, what is the maximum possible daily output of machine B?

Question 2

Two weekly collections are in the ratio 2 : 9 . If k is the common natural-number multiplier, which pair correctly represents the first collection and the second collection?

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