XAT Number Systems MCQs with Solutions: Practice XAT Previous Year Questions (PYQs)

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Number Systems is an important topic in the Quantitative Ability and Data Interpretation section in XAT exam. Practising this topic will increase your score overall and make your conceptual grip on XAT exam stronger.

This article gives you a full set of XAT Number Systems MCQs with explanations and XAT previous year questions (PYQs) for effective practice. Practice of Quantitative Ability and Data Interpretation MCQs including Number Systems questions regularly will improve accuracy, speed, and confidence in the XAT 2026 exam.

Also Check : XAT 2026 Question Paper with Solutions PDF

XAT Number Systems MCQs with Solutions

1.
An archer, from the point A, is aiming at a target, placed on the top of a 56 feet tall tree. The base of the tree is at the point C. From the archer's position, the angle of elevation to the target is 45° from his eye level. The archer, facing the tree, moves backwards on the straight line joining the points A and C, to a new position at the point B. From the point B, the angle of elevation from his eye level to the target becomes 30°.
How far did the archer move from A to B (in feet) if his eye level is at a height of 6 feet from the ground?
- \( \frac{50}{\sqrt{3}} \)
- \( 50 \sqrt{3} \)
- \( 56 \sqrt{3} \)
- \( 50(\sqrt{3} - 1) \)
- \( 50(1 - \frac{1}{\sqrt{3}}) \)
View Solution
2.
The roots of the polynomial are the radii of three concentric circles \(P(x) = 2x^3-11x^2 +17x-6\). The ratio of their area, when arranged from the largest to the smallest, is:
- 6:2:1
- 9:4:1
- 16:6:3
- 36:16:1
- None of the remaining options is correct.
View Solution
3.
A king has distributed all his rare jewels in three boxes. The first box contains 1/3 of the rare jewels, while the second box contains k/5 of the rare jewels, for some positive integer value of k. The third box contains 66 rare jewels.
How many rare jewels does the king have?
- 990
- 660
- 240
- 1080
- Cannot be determined uniquely from the given information.
View Solution
4.
Consider an+1 = 1/1+1/an for n = 1,2,.......2008,2009. where a1 = 1. Find the value of a1a2 + a2a3 + a3a4+ .......+ a2008a2009.
- 2009/1000
- 2009/2008
- 2008/2009
- 6000/2009
- 2008/6000
View Solution
5.
How many solutions \( (x, y, z) \) of the equation \( x + y^2 + z^3 = 50 \) exist, where \( x, y \) and \( z \) are positive integers?
- 15
- 18
- 16
- 17
- None of the other options is correct
View Solution
6.
The least common multiple of a number and 990 is 6930. The greatest common divisor of that number and 550 is 110.
What is the sum of the digits of the least possible value of that number?
- 6
- 9
- 14
- 18
- None of the remaining options is correct
View Solution
7.
Consider the real-valued function \(f(x) = \log \left( \frac{3x - 7}{\sqrt{2x \times 2x}} \right) - 7x + f\). Find the domain of f(x).
- (-∞, 7/3)
- R - (3/2, 2)
- (7/3, ∞)
- R - {3/2, 2, 7/3}
- R - {7/3}
View Solution
8.
Arun selected an integer \( x \) between 2 and 40, both inclusive. He noticed that the greatest common divisor of the selected integer \( x \) and any other integer between 2 and 40, both inclusive, is 1. How many different choices for such an \( x \) are possible?
- 1
- 8
- 4
- 0
- 12
View Solution
9.
Suppose Haruka has a special key Δ in her calculator called delta key:
Rule 1: If the display shows a one-digit number, pressing delta key
Δ replaces the displayed number with twice its value.
Rule 2: If the display shows a two-digit number, pressing delta key
Δ replaces the displayed number with the sum of the two digits.
Suppose Haruka enters the value 1 and then presses delta key
Δ repeatedly. After pressing the Δ key for 68 times, what will be the displayed number?
- 7
- 4
- 10
- 2
- 8
View Solution
10.
Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be a real-valued function defined as \( f(0, y) = y + 1 \) and \( f(x + 1, y) = f(x, f(x, y)) + x \).
What is the value of \( f(2, 2) \)?
- 6
- 5
- None of the other options is correct
- 7
- 4
View Solution
11.
Consider a 4-digit number of the form abbb, i.e., the first digit is a (a > 0) and the last three digits are all b.
Which of the following conditions is both NECESSARY and SUFFICIENT to ensure that the 4- digit number is divisible by a?
- b is divisible by a
- b is equal to 0
- 21b is divisible by a
- 9b is divisible by a
- 3b is divisible by a
View Solution
12.
A supplier receives orders from 5 different buyers. Each buyer places their order only on a Monday. The first buyer places the order after every 2 weeks, the second buyer, after every 6 weeks, the third buyer, after every 8 weeks, the fourth buyer, every 4 weeks, and the fifth buyer, after every 3 weeks. It is known that on January 1st, which was a Monday, each of these five buyers placed an order with the supplier.
On how many occasions, in the same year, will these buyers place their orders together excluding the order placed on January 1st?
- 1
- 5
- 4
- 2
- 3
View Solution
13.
Three categories of candidates appear for an admission test: diligent (10%), lazy (30%) and confused (60%). A diligent candidate is 10 times more likely to clear the admission test compared to a lazy candidate.
If 40% of the candidates clearing the admission test are confused, what is the MAXIMUM possible value of the probability of a confused candidate clearing the test?
- \( \frac{13}{37} \)
- \( \frac{6}{7} \)
- \( \frac{13}{90} \)
- \( \frac{37}{100} \)
- \( \frac{2}{3} \)
View Solution
14.
The sum of the cubes of two numbers is 128, while the sum of the reciprocals of their cubes is b. What is the product of the squares of the numbers?
- 64
- 16
- 256
- 48
- 32
View Solution
15.
The Madhura Fruits Company is packing four types of fruits into boxes. There are 126 oranges, 162 apples, 198 guavas and 306 pears. The fruits must be packed in such a way that a given box must have only one type of fruit and must contain the same number of fruit units as any other box. What is the minimum number of boxes that must be used?
- 21
- 42
- 36
- 18
- 44
View Solution

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XAT Number Systems MCQs with Solutions: Practice XAT Previous Year Questions (PYQs)

XAT Number Systems MCQs with Solutions

2.
The roots of the polynomial are the radii of three concentric circles \(P(x) = 2x^3-11x^2 +17x-6\). The ratio of their area, when arranged from the largest to the smallest, is:

3.
A king has distributed all his rare jewels in three boxes. The first box contains 1/3 of the rare jewels, while the second box contains k/5 of the rare jewels, for some positive integer value of k. The third box contains 66 rare jewels.
How many rare jewels does the king have?

4.
Consider an+1 = 1/1+1/an for n = 1,2,.......2008,2009. where a1 = 1. Find the value of a1a2 + a2a3 + a3a4+ .......+ a2008a2009.

5.
How many solutions \( (x, y, z) \) of the equation \( x + y^2 + z^3 = 50 \) exist, where \( x, y \) and \( z \) are positive integers?

6.
The least common multiple of a number and 990 is 6930. The greatest common divisor of that number and 550 is 110.
What is the sum of the digits of the least possible value of that number?

7.
Consider the real-valued function \(f(x) = \log \left( \frac{3x - 7}{\sqrt{2x \times 2x}} \right) - 7x + f\). Find the domain of f(x).

8.
Arun selected an integer \( x \) between 2 and 40, both inclusive. He noticed that the greatest common divisor of the selected integer \( x \) and any other integer between 2 and 40, both inclusive, is 1. How many different choices for such an \( x \) are possible?

10.
Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be a real-valued function defined as \( f(0, y) = y + 1 \) and \( f(x + 1, y) = f(x, f(x, y)) + x \).
What is the value of \( f(2, 2) \)?

11.
Consider a 4-digit number of the form abbb, i.e., the first digit is a (a > 0) and the last three digits are all b.
Which of the following conditions is both NECESSARY and SUFFICIENT to ensure that the 4- digit number is divisible by a?

14.
The sum of the cubes of two numbers is 128, while the sum of the reciprocals of their cubes is b. What is the product of the squares of the numbers?

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XAT Number Systems MCQs with Solutions

2.The roots of the polynomial are the radii of three concentric circles \(P(x) = 2x^3-11x^2 +17x-6\). The ratio of their area, when arranged from the largest to the smallest, is:

3.A king has distributed all his rare jewels in three boxes. The first box contains 1/3 of the rare jewels, while the second box contains k/5 of the rare jewels, for some positive integer value of k. The third box contains 66 rare jewels. How many rare jewels does the king have?

4.Consider an+1 = 1/1+1/an for n = 1,2,.......2008,2009. where a1 = 1. Find the value of a1a2 + a2a3 + a3a4+ .......+ a2008a2009.

5.How many solutions \( (x, y, z) \) of the equation \( x + y^2 + z^3 = 50 \) exist, where \( x, y \) and \( z \) are positive integers?

6.The least common multiple of a number and 990 is 6930. The greatest common divisor of that number and 550 is 110. What is the sum of the digits of the least possible value of that number?

7.Consider the real-valued function \(f(x) = \log \left( \frac{3x - 7}{\sqrt{2x \times 2x}} \right) - 7x + f\). Find the domain of f(x).

8.Arun selected an integer \( x \) between 2 and 40, both inclusive. He noticed that the greatest common divisor of the selected integer \( x \) and any other integer between 2 and 40, both inclusive, is 1. How many different choices for such an \( x \) are possible?

10.Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be a real-valued function defined as \( f(0, y) = y + 1 \) and \( f(x + 1, y) = f(x, f(x, y)) + x \). What is the value of \( f(2, 2) \)?

11.Consider a 4-digit number of the form abbb, i.e., the first digit is a (a > 0) and the last three digits are all b. Which of the following conditions is both NECESSARY and SUFFICIENT to ensure that the 4- digit number is divisible by a?

14.The sum of the cubes of two numbers is 128, while the sum of the reciprocals of their cubes is b. What is the product of the squares of the numbers?

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2.
The roots of the polynomial are the radii of three concentric circles \(P(x) = 2x^3-11x^2 +17x-6\). The ratio of their area, when arranged from the largest to the smallest, is:

3.
A king has distributed all his rare jewels in three boxes. The first box contains 1/3 of the rare jewels, while the second box contains k/5 of the rare jewels, for some positive integer value of k. The third box contains 66 rare jewels.
How many rare jewels does the king have?

4.
Consider an+1 = 1/1+1/an for n = 1,2,.......2008,2009. where a1 = 1. Find the value of a1a2 + a2a3 + a3a4+ .......+ a2008a2009.

5.
How many solutions \( (x, y, z) \) of the equation \( x + y^2 + z^3 = 50 \) exist, where \( x, y \) and \( z \) are positive integers?

6.
The least common multiple of a number and 990 is 6930. The greatest common divisor of that number and 550 is 110.
What is the sum of the digits of the least possible value of that number?

7.
Consider the real-valued function \(f(x) = \log \left( \frac{3x - 7}{\sqrt{2x \times 2x}} \right) - 7x + f\). Find the domain of f(x).

8.
Arun selected an integer \( x \) between 2 and 40, both inclusive. He noticed that the greatest common divisor of the selected integer \( x \) and any other integer between 2 and 40, both inclusive, is 1. How many different choices for such an \( x \) are possible?

10.
Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be a real-valued function defined as \( f(0, y) = y + 1 \) and \( f(x + 1, y) = f(x, f(x, y)) + x \).
What is the value of \( f(2, 2) \)?

11.
Consider a 4-digit number of the form abbb, i.e., the first digit is a (a > 0) and the last three digits are all b.
Which of the following conditions is both NECESSARY and SUFFICIENT to ensure that the 4- digit number is divisible by a?

14.
The sum of the cubes of two numbers is 128, while the sum of the reciprocals of their cubes is b. What is the product of the squares of the numbers?