Number System for UPSC CSAT – Topics & PYQs Practice
Feb, 2026
•5 min read
CSAT can feel unpredictable for many aspirants. But if there is one area you can prepare in a clear, structured, and scoring way, it is the Number System.
Every year, around 15–20% of the CSAT paper comes from this topic. More importantly, it is the base for many other maths topics like percentages, ratios, averages, time & work, and profit & loss. If your number basics are strong, these topics become much easier and solving questions becomes faster and more accurate.
Here, you will understand important concepts of the Number System, along with useful strategies and CSAT Previous Year Questions (PYQs), so that you can prepare this topic in a structured way. Let's get started!
Types of Numbers
In mathematics, numbers are grouped based on their properties. Understanding these groups clearly will help you avoid confusion in basic and conceptual questions in CSAT.
| Type of Number | Symbol | Definition | Examples | Important Note |
|---|---|---|---|---|
| Natural Numbers | N | Positive numbers used for counting objects. They start from 1 and go up infinitely. | 1, 2, 3, 4… | 0 and negative numbers are not included. |
| Whole Numbers | W | All natural numbers along with 0. | 0, 1, 2, 3… | No negative numbers. |
| Integers | Z | All positive and negative whole numbers, including 0. | -3, -2, -1, 0, 1, 2… | Fractions and decimals are not included. |
| Rational Numbers | Q | Numbers that can be written in the form p/q, where p and q are integers and q ≠ 0. | 1/2, -4/3, 5, 0.75 | Includes terminating and repeating decimals. |
| Irrational Numbers | — | Numbers that cannot be written in the form p/q. Their decimal form is non-terminating and non-repeating. | π, √2, √5 | Cannot be expressed as exact fractions. |
| Real Numbers | R | All rational and irrational numbers combined. These are all numbers that lie on the number line. | -5, 0, 3/4, √3, π | Includes almost every number used in basic mathematics. |
Also see: UPSC CSAT Syllabus 2026: Complete Preparation Guide with Topper Strategies
Key Concepts in Number System
In the Number System, certain core concepts are repeatedly tested in the UPSC Prelims CSAT. If these fundamentals are clear, solving questions becomes much easier and faster. Here is a list of the most important concepts:
1. Factors and Multiples
- A factor of a number divides it exactly.
- A multiple is the result of multiplying a number by an integer.
2. Prime and Composite Numbers
- A prime number has exactly two factors: 1 and itself.
- A composite number has more than two factors.
3. Divisibility Rules
- Rules that help determine whether a number is divisible by another number without actual division.
4. LCM (Least Common Multiple)
- The smallest number that is a multiple of two or more numbers.
5. HCF / GCD (Highest Common Factor / Greatest Common Divisor)
- The largest number that divides two or more numbers exactly.
6. Remainder Theorem Concept
- Helps determine the remainder when one number is divided by another, often using patterns or cyclicity.
7. Even and Odd Numbers
- Even numbers are divisible by 2.
- Odd numbers are not divisible by 2.
8. Co-prime Numbers
- Two numbers whose HCF is 1.
9. Cyclicity of Numbers
- The repeating pattern of digits in powers of certain numbers.
10. Base System (Number Representation)
- Writing numbers in different bases, such as binary (base 2), decimal (base 10), etc.
Now, let's solve UPSC CSAT PYQs based on the above concepts with easy explanations and conceptual understanding.
UPSC Prelims CSAT PYQs on Number System
QUESTION 1
CSAT
Easy
Maths
Prelims 2019
The ratio of a two-digit natural number to a number formed by reversing its digits is . The number of such pairs is
Select an option to attempt
QUESTION 2
CSAT
Easy
Maths
Prelims 2023
Consider a 3-digit number.
Question: What is the number? Statement-1: The sum of the digits of the number is equal to the product of the digits. Statement-2: The number is divisible by the sum of the digits of the number.
Which one of the following is correct in respect of the above Question and the Statements?
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QUESTION 3
CSAT
Hard
Maths
Prelims 2025
There are n sets of numbers each having only three positive integers with LCM equal to 1001 and HCF equal to 1. What is the value of n?
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QUESTION 4
CSAT
Medium
Maths
Prelims 2025
Consider the first 100 natural numbers. How many of them are not divisible by any one of , , , , and ?
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QUESTION 5
CSAT
Easy
Maths
Prelims 2023
What is the remainder when is divided by 100?
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QUESTION 6
CSAT
Medium
Maths
Prelims 2025
A 4-digit number N is such that when divided by 3, 5, 6, and 9 it leaves a remainder of 1, 3, 4, and 7 respectively. What is the smallest value of N?
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QUESTION 7
CSAT
Easy
Maths
Prelims 2022
How many 3-digit natural numbers (without repetition of digits) are there such that each digit is odd and the number is divisible by 5?
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QUESTION 8
CSAT
Easy
Maths
Prelims 2021
The number is divisible by 7. What is the value of the digit P?
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QUESTION 9
CSAT
Hard
Maths
Prelims 2020
How many integers are there between 1 and 100 which have 4 as a digit but are not divisible by 4?
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QUESTION 10
CSAT
Medium
Maths
Prelims 2018
X and Y are natural numbers other than 1, and Y is greater than X. Which of the following represents the largest number?
Select an option to attempt
Practice UPSC Prelims CSAT PYQs (2013-2025)
Do interactive practise of UPSC CSAT PYQs year-wise, covering Comprehension, Reasoning, and Quantitative Aptitude, with detailed explanations.
Start Practicing NowHow to Prepare Number System Effectively for CSAT
Preparing Number System does not require advanced mathematics. It requires clear basics, regular practice, and smart revision. If you follow a structured approach, this topic can become one of your strongest scoring areas. Here is a practical way to prepare:
1. Strengthen Your Fundamentals First
- Start with basic definitions and concepts — factors, multiples, primes, divisibility rules, LCM, HCF, and remainders.
- Do not rush to shortcuts before understanding the logic.
2. Master Divisibility Rules & Factorisation
- Divisibility rules (especially for 2, 3, 4, 5, 6, 8, 9, 11) save a lot of time in the exam.
- Practice breaking numbers into prime factors mentally. This improves speed in LCM, HCF, and remainder-based questions.
3. Practice Remainders & Cyclicity Deeply
- UPSC frequently asks questions based on powers and remainders. Instead of calculating large powers, learn to identify repeating patterns (cycles).
- Memorise cyclicity patterns of last digits (like powers of 2, 3, 7, 8).
4. Solve Previous Year Questions (PYQs)
- PYQs help you understand the level and pattern of questions asked by UPSC.
- After understanding the concepts, practice at least the last 10 years of CSAT questions from Number System.
5. Improve Calculation Speed with Discipline
CSAT is qualifying, but you cannot afford careless mistakes.
- Avoid lengthy calculations when logic can be used.
- Use option elimination wherever possible.
- Check signs (+/−) carefully in integer questions.
6. Revise with Short Notes
- Prepare a one-page revision sheet of Number System formulas, properties, and important tricks.
- Revise it regularly to keep concepts fresh.
If you prepare Number System in this structured way, you will surely improve your confidence in mathematics.
Must see: All Important UPSC CSAT Formulas for Prelims 2026 – Quick Revision for Prelims
Final Words
Number System is not a difficulty if your basics are strong. Do not just read the concepts. Practice questions regularly. Solve PYQs multiple times. Identify your mistakes and correct them. With steady effort, your speed and accuracy will automatically improve.
CSAT is qualifying, but it has eliminated many serious aspirants. Prepare smartly, practice consistently, and turn CSAT into your strength.
Must see: Coding Decoding Questions and Answers: A Guide for UPSC CSAT
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