Number Theory and Pattern Puzzles Previous Year Questions (PYQs) for XAT: 2+ Solved Questions with Step-by-Step Solutions

Solve 2+ Number Theory and Pattern Puzzles previous year questions for XAT with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Number Theory & Pattern Puzzles

Chapter Roadmap

Divisibility and Factors
The DNA of numbers. Prime factorization, counting factors, and sum of factors.

HCF and LCM
The intersection and union of factors. Translating word problems into mathematical logic.

Remainders and Cyclicity
Finding the last digits and remainders of massive powers using cyclic patterns.

Factorials and Trailing Zeros
Legendre's formula and counting the hidden tens in large products.

Pattern Puzzles
Decoding recursive sequences, cyclic operations, and logical constraints.

The Intuition: Integers as Lego Blocks

The Intuition

The Core Philosophy: Every integer greater than 1 is either a prime (a single, unbreakable block) or a composite (a structure built from prime blocks).

Why this matters for exams:

Sharing blocks: This is the Highest Common Factor (HCF).

Combining all blocks: This is the Least Common Multiple (LCM).

Leftover blocks: This is the Remainder.

Number Theory and Pattern Puzzles: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Aptitude and Data Interpretation (QA & DI) MCQ

The addition of 7 distinct positive integers is 1740. What is the largest possible “greatest common divisor” of these 7 distinct positive integers?

A.
42
B.
60
C.
74
D.
140
E.
None of the above.

Correct Answer:

Step-by-Step Solution

Key idea: This is a Maximizing GCD with Fixed Sum question. We need to maximize the GCD $g$ of 7 distinct positive integers $a_{1}, \dots, a_{7}$ such that $\sum a_{i} = 1740$ .

Step 1: Express the integers in terms of their GCD.

Let the GCD be $g$ . Then each integer $a_{i}$ can be written as $a_{i} = g \cdot k_{i}$ , where $k_{i}$ are distinct positive integers.

Since $g$ is the greatest common divisor, $g cd (k_{1}, \dots, k_{7}) = 1$ . (Though for maximization, we primarily care about the sum constraint first).

Step 2: Use the sum constraint.

$\sum_{i = 1}^{7} a_{i} = 1740$

$\sum_{i = 1}^{7} g \cdot k_{i} = 1740$

$g \cdot (\sum_{i = 1}^{7} k_{i}) = 1740$

Let $S_{k} = \sum_{i = 1}^{7} k_{i}$ .

Then $g \cdot S_{k} = 1740$ .

This implies $g$ must be a divisor of 1740.

Step 3: Minimize $S_{k}$ to maximize $g$ .

To maximize $g$ , we must minimize $S_{k}$ .

The $k_{i}$ are distinct positive integers. To minimize their sum, we should choose the smallest possible distinct positive integers: $1, 2, 3, 4, 5, 6, 7$ .

Minimum $S_{k} = 1 + 2 + 3 + 4 + 5 + 6 + 7 = \frac{7 \times 8}{2} = 28$ .

So, $S_{k} \geq 28$ .

Consequently, $g = \frac{1740}{S _{k}} \leq \frac{1740}{28}$ .

Step 4: Calculate the upper bound for $g$ .

$1740/28 \approx 62.14$ .

So, $g \leq 62$ .

Step 5: Check divisors of 1740 less than or equal to 62.

First, factorize 1740.

$1740 = 10 \times 174 = 10 \times 2 \times 87 = 20 \times 3 \times 29 = 2^{2} \cdot 3 \cdot 5 \cdot 29$ .

We need a divisor $g$ of 1740 such that $g \leq 62$ and $S_{k} = 1740/ g$ can be formed by sum of 7 distinct positive integers.

Note: If we pick a specific $g$ , $S_{k}$ is fixed. We must ensure there exist 7 distinct positive integers summing to $S_{k}$ .

The condition for existence is simply $S_{k} \geq 28$ . (Since any sum $\geq 28$ can be formed by distinct positive integers, e.g., start with 1..7 and add the excess to the largest number).

So we just need the largest divisor of 1740 that is $\leq 62$ .

Let's list divisors of 1740 near 62.

Divisors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 29, 30, ...

Next ones:

$1740/29 = 60$ . So 60 is a divisor.

$1740/30 = 58$ . So 58 is a divisor? No, $1740/58 = 30$ . Yes.

Is there a divisor between 60 and 62?

Check 61: Prime? $1740/61 \approx 28.5$ . No.

Check 62: $1740/62 \approx 28.06$ . No.

So the largest divisor $\leq 62$ is 60.

Step 6: Verify if $g = 60$ works.

If $g = 60$ , then $S_{k} = 1740/60 = 29$ .

Can we find 7 distinct positive integers summing to 29?

Min sum is 28 ( $1 + 2 + 3 + 4 + 5 + 6 + 7$ ).

We need sum 29. Just increase the largest term by 1: $1, 2, 3, 4, 5, 6, 8$ .

Sum = 29. Distinct? Yes.

$g cd (1, 2, 3, 4, 5, 6, 8) = 1$ .

So the numbers are $60, 120, 180, 240, 300, 360, 480$ .

GCD is 60. Sum is $60 (29) = 1740$ .

Answer: 60.

Question 2 · Quantitative Aptitude and Data Interpretation (QA & DI) MCQ

Suppose Haruka has a special key

△

in her caculator called delta key:
Rule 1: If the display shows a one-digit number, pressing delta key

△

replace the displayed number with twice its value.
Rule 2: If the display shows a two-digits number, pressing delta key

△

replace the displayed number with the number sum of two digits.
Suppose Haruka enters the value 1 and then presses delta key

△

repeated.
After pressing the key for 68 times, what will be the displayed number?

A.
7
B.
4
C.
10
D.
2
E.
8

Correct Answer:

Step-by-Step Solution

Key idea: This is a Pattern Puzzle / Cyclic Operation question. We need to simulate the operation until a cycle is detected, then use modular arithmetic to find the state after 68 steps.

Step 1: Simulate the first few steps starting from 1.

Start: 1

Press 1: Rule 1 (1-digit) $\to 1 \times 2 = 2$ . Display: 2

Press 2: Rule 1 (1-digit) $\to 2 \times 2 = 4$ . Display: 4

Press 3: Rule 1 (1-digit) $\to 4 \times 2 = 8$ . Display: 8

Press 4: Rule 1 (1-digit) $\to 8 \times 2 = 16$ . Display: 16

Press 5: Rule 2 (2-digits) $\to 1 + 6 = 7$ . Display: 7

Press 6: Rule 1 (1-digit) $\to 7 \times 2 = 14$ . Display: 14

Press 7: Rule 2 (2-digits) $\to 1 + 4 = 5$ . Display: 5

Press 8: Rule 1 (1-digit) $\to 5 \times 2 = 10$ . Display: 10

Press 9: Rule 2 (2-digits) $\to 1 + 0 = 1$ . Display: 1

Step 2: Identify the cycle.

After 9 presses, we are back to 1.

The sequence of displayed numbers after each press is:

$n = 1 : 2$

$n = 2 : 4$

$n = 3 : 8$

$n = 4 : 16$

$n = 5 : 7$

$n = 6 : 14$

$n = 7 : 5$

$n = 8 : 10$

$n = 9 : 1$

$n = 10 : 2$ (Cycle repeats)

The cycle length is 9. The values repeat every 9 presses.

The sequence is: $2, 4, 8, 16, 7, 14, 5, 10, 1$ .

Step 3: Calculate the position in the cycle for 68 presses.

We need the value after 68 presses.

Find $68 (mod 9)$ .

$68 = 7 \times 9 + 5$ .

Remainder is 5.

This means the 68th press results in the same value as the 5th press.

Step 4: Retrieve the 5th value.

From our list:

1st: 2

2nd: 4

3rd: 8

4th: 16

5th: 7

Wait, let me re-check the options. Options are 7, 4, 10, 2, 8.

My calculated answer is 7. Option A is 7.

Let me re-read the question carefully.

"Haruka enters the value 1 and then presses delta key repeated. After pressing the key for 68 times..."

Let's re-simulate carefully.

Initial: 1

Press 1: $1 \to 2$

Press 2: $2 \to 4$

Press 3: $4 \to 8$

Press 4: $8 \to 16$

Press 5: $16 \to 1 + 6 = 7$

Press 6: $7 \to 14$

Press 7: $14 \to 1 + 4 = 5$

Press 8: $5 \to 10$

Press 9: $10 \to 1 + 0 = 1$

Press 10: $1 \to 2$

Cycle is indeed 9.

$68 \equiv 5 (mod 9)$ .

The 5th term is 7.

Why is the provided current_answer null? And why did I think B?

Let's check the options again.

A: 7

B: 4

C: 10

D: 2

E: 8

My calculation gives 7. So Answer is A.

Let me double check the "Sum of two digits" rule.

"If the display shows a two-digits number... replace... with the number sum of two digits."

Yes, $16 \to 7$ . $14 \to 5$ . $10 \to 1$ .

Is it possible the cycle starts later?

Sequence: 2, 4, 8, 16, 7, 14, 5, 10, 1, 2...

It returns to 1, which was the initial state before any press.

So the state after press 9 is 1.

The state after press 10 is 2.

So $V a l u e (n) = V a l u e (n - 9)$ .

$V a l u e (68) = V a l u e (68 - 9) = \dots = V a l u e (5)$ .

$V a l u e (5) = 7$ .

Answer is 7.

Wait, looking at the options provided in the prompt JSON:

"options": ["7", "4", "10", "2", "8"]

If the answer is 7, it corresponds to the first option.

Let me check if I made an arithmetic error.

$1 \times 2 2$

$2 \times 2 4$

$4 \times 2 8$

$8 \times 2 16$

$16 + 7$

$7 \times 2 14$

$14 + 5$

$5 \times 2 10$

$10 + 1$

Cycle: 2, 4, 8, 16, 7, 14, 5, 10, 1. Length 9.

$68/9 = 7$ rem $5$ .

5th item is 7.

Answer is 7.

Number Theory and Pattern Puzzles Previous Year Questions (PYQs) for XAT: 2+ Solved Questions with Step-by-Step Solutions

Chapter Roadmap: Number Theory & Pattern Puzzles

The Intuition: Integers as Lego Blocks

Number Theory and Pattern Puzzles: Solved Questions with Step-by-Step Explanations (2 Problems)

Step-by-Step Solution

Step-by-Step Solution

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