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

Number Theory Word Problems and Diophantine Conditions short notes for CAT: 16 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

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?

The Core Move: Isolate One Variable as a Fraction

Core CAT Move

When variables must be natural numbers, isolate one variable:

variable = \frac{integer expression}{fixed divisor}

Then ask:

Is it positive? Is it an integer? Does it satisfy restrictions?

Example shape:

b = \frac{a ( 3 c - 8 )}{8}

For

b

to be a natural number,

a (3 c - 8)

must be divisible by

8

, and

3 c - 8

must be positive.

Bounding: Why Small Values Matter First

When Asked: “Smallest Possible Value”

If the target is

3 a + 2 b + c,

then large

a

b

, or

c

quickly increases the answer.

Find the minimum allowed range
For example, if

3 c - 8 > 0

, then

c \geq 3

Test small cases with divisibility
Do not test randomly; use the denominator condition.

Stop only after you can justify no smaller case works
Small-looking values may fail because of natural-number or distinctness conditions.

CAT PYQ: Compute and Confirm the Minimum

Evaluate $3 a + 2 b + c$

Candidate	Distinct?	Value of $3 a + 2 b + c$
$(8, 1, 3)$	Yes	$24 + 2 + 3 = 29$
$(2, 1, 4)$	Yes	$6 + 2 + 4 = 12$
$(8, 7, 5)$	Yes	$24 + 14 + 5 = 43$

Final PYQ Answer

12

Achieved by $a = 2, b = 1, c = 4$ .

Check in original equation:

3 a c = 3 \cdot 2 \cdot 4 = 24

and

8 (a + b) = 8 (2 + 1) = 24

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 short notes in this unit

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

Number Theory Word Problems and Diophantine Conditions short notes for CAT: 16 study cards covering concepts, formulas, shortcuts and exam traps, plus solved

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

Natural Numbers Are Not Just Decoration

Natural Number Constraint

The Core Move: Isolate One Variable as a Fraction

Core CAT Move

Bounding: Why Small Values Matter First

When Asked: “Smallest Possible Value”

CAT PYQ: Compute and Confirm the Minimum

Evaluate 3a+2b+c

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

Step-by-Step Solution

Step-by-Step Solution

More short notes in this unit

A question from this chapter

Login to view the complete short notes

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Evaluate $3 a + 2 b + c$