UPSC Prelims CSAT: Divisibility Rules and PYQs Practice
Master divisibility rules for UPSC CSAT with easy examples, PYQs, shortcuts, and practice questions. Improve calculation speed, accuracy, and boost your Prelims score.
Jul, 2026
•6 min read
Divisibility Rules are one of the most important topics in UPSC Prelims CSAT Quantitative Aptitude. They help you perform calculations quickly, eliminate incorrect options, and solve aptitude questions with greater accuracy.
Here, you will learn all the important divisibility rules for UPSC CSAT, understand them through examples, and practise UPSC CSAT Previous Year Questions (PYQs). This guide will help you strengthen your number system concepts and approach CSAT questions with confidence.
What is Divisibility?
Divisibility refers to the ability of one number to be divided by another without leaving any remainder. If a number is completely divisible by another number, the remainder is 0.
For example:
- 24 ÷ 6 = 4 (Remainder = 0), so 24 is divisible by 6.
- 25 ÷ 6 = 4 (Remainder = 1), so 25 is not divisible by 6.
In the UPSC CSAT, divisibility helps you solve questions on Number System, Simplification, HCF and LCM, Factors and Multiples, and Quantitative Aptitude more efficiently. Instead of performing lengthy calculations, you can apply divisibility rules to quickly identify the correct answer, eliminate wrong options, and save valuable time in the examination.
Divisibility Rules for UPSC CSAT
Learning the divisibility rules can help you solve UPSC CSAT Quantitative Aptitude questions much faster. These shortcuts reduce lengthy calculations, improve accuracy, and make it easier to eliminate incorrect options in the examination.
| Number | Divisibility Rule | Example |
|---|---|---|
| 2 | A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8). | 246 → Last digit is 6 ✓ |
| 3 | A number is divisible by 3 if the sum of its digits is divisible by 3. | 237 → 2 + 3 + 7 = 12 ✓ |
| 4 | A number is divisible by 4 if its last two digits are divisible by 4. | 516 → 16 ÷ 4 = 4 ✓ |
| 5 | A number is divisible by 5 if its last digit is 0 or 5. | 135 → Last digit is 5 ✓ |
| 7 | Double the last digit and subtract it from the remaining number. Repeat if needed. If the result is divisible by 7, so is the original number. | 672 → 67 − (2 × 2) = 63 ✓ |
| 8 | A number is divisible by 8 if its last three digits are divisible by 8. | 5,192 → 192 ÷ 8 = 24 ✓ |
| 9 | A number is divisible by 9 if the sum of its digits is divisible by 9. | 288 → 2 + 8 + 8 = 18 ✓ |
| 10 | A number is divisible by 10 if its last digit is 0. | 430 → Last digit is 0 ✓ |
| 11 | Find the difference between the sum of digits in alternate positions. If the difference is 0 or divisible by 11, the number is divisible by 11. | 308 → (3 + 8) − 0 = 11 ✓ |
| 13 | Multiply the last digit by 9, then add it to the remaining digits. Repeat if required. If the result is divisible by 13, so is the original number. | 247 → 24 + (7 × 9) = 87 → 87 ÷ 13 ✓ |
| 17 | Multiply the last digit by 5 and subtract it from the remaining number. Repeat if needed. If the result is divisible by 17, so is the original number. | 221 → 22 − (1 × 5) = 17 ✓ |
Mentor Tip: In the UPSC CSAT, divisibility rules are powerful shortcuts. Use them to eliminate options quickly, verify answers without lengthy calculations, and solve Number System, HCF & LCM, Simplification, and Data Sufficiency questions more efficiently.
How to Use Divisibility Rules in UPSC CSAT
To score well in the CSAT paper, you must know where and how to apply them. These rules help you solve questions faster, reduce unnecessary calculations, and improve your accuracy. You can use divisibility rules in the following types of CSAT questions:
1. Number System Questions: Check whether a number is divisible by another number without performing long division.
Example: Is 5,436 divisible by 9?
Solution: 5 + 4 + 3 + 6 = 18, which is divisible by 9. Therefore, 5,436 is divisible by 9.
2. HCF and LCM Problems: Quickly identify the factors of a number using divisibility rules before finding the Highest Common Factor (HCF) or Least Common Multiple (LCM).
3. Simplification Questions: Reduce fractions and simplify numerical expressions by checking divisibility instead of performing lengthy calculations.
4. Data Sufficiency Questions: Use divisibility rules to determine whether the given information is enough to answer the question without solving it completely.
5. Option Elimination: Instead of calculating every option, apply divisibility rules to eliminate incorrect choices and arrive at the correct answer more quickly.
Tip: CSAT paper often tests your logical approach rather than lengthy calculations. Memorising and applying divisibility rules can save valuable time, especially when solving Quantitative Aptitude questions under pressure.
Download the PDF: UPSC Prelims 2026 CSAT Paper II
UPSC Prelims CSAT Divisibility PYQs with Solutions
QUESTION 1
If and are two digits and the number is divisible by , then what is the remainder, if is divided by ?
QUESTION 2
What is the maximum value of n such that is divisible by ?
QUESTION 3
For any choices of values of X, Y and Z, the 6 digit number of the form is divisible by:
QUESTION 4
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?
QUESTION 5
is divisible by
QUESTION 6
If is divided by 10, then what is the remainder?
QUESTION 7
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?
QUESTION 8
What is the least four-digit number when divided by 3, 4, 5 and 6 leaves a remainder 2 in each case?
QUESTION 9
If , where is not divisible by , then the value of is
QUESTION 10
Certain 3-digit numbers have the following characteristics:
- All the three digits are different.
- The number is divisible by 7.
- The number on reversing the digits is also divisible by 7.
How many such 3-digit numbers are there?
Conclusion
Mastering divisibility rules for UPSC CSAT can significantly improve your speed, accuracy, and confidence in the Prelims examination. These simple shortcuts help you solve Number System, HCF & LCM, Simplification, and Data Sufficiency questions with fewer calculations and greater precision. Revise these rules regularly, practise UPSC CSAT PYQs, and apply them consistently to maximise your CSAT score.
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