CDS & AFCAT 1 2025 Exam Maths Number System Class 2

A recent class was conducted on the topic of Number System as part of the preparation for the Combined Defence Services (CDS) and Air Force Common Admission Test (AFCAT) Exams. The session focused on key subtopics, particularly Divisibility Rules for numbers like 11, 12, 13, 16, 17, 25, and 125, and methods for remainder finding using theorems and techniques for finding the unit digit of large powers. This blog will discuss these topics in detail, along with strategies to help you prepare efficiently for the upcoming exams.

The Number System: An Essential Topic

The number system forms the foundation of quantitative aptitude, not just in competitive exams like CDS and AFCAT but in all exams that test mathematical skills. This topic covers various aspects such as types of numbers (natural, whole, integers, rational, irrational), prime and composite numbers, divisibility rules, finding remainders, and unit digit problems. The ability to solve problems related to the number system efficiently can give you an edge in these exams.

Divisibility Rules: Simplifying Large Numbers

Divisibility rules help us quickly determine whether a number can be divided by another number without performing the actual division. For the purpose of the CDS and AFCAT exams, understanding the divisibility rules for 11, 12, 13, 16, 17, 25, and 125 is crucial. Let’s break them down:

1. Divisibility by 11: To check if a number is divisible by 11, find the difference between the sum of its digits in odd positions and the sum of its digits in even positions. If the result is either 0 or divisible by 11, then the number is divisible by 11.
2. Divisibility by 12: A number is divisible by 12 if it is divisible by both 3 and 4. So, check the divisibility by 3 (sum of digits divisible by 3) and divisibility by 4 (the last two digits of the number divisible by 4).
3. Divisibility by 13: The rule for 13 is more complex. One approach is to multiply the last digit of the number by 4 and subtract it from the remaining part of the number. Repeat the process if needed, and if the result is divisible by 13, the original number is also divisible by 13.
4. Divisibility by 16: For divisibility by 16, check if the last four digits of the number are divisible by 16. If yes, the entire number is divisible by 16.
5. Divisibility by 17: Similar to the rule for 13, multiply the last digit by 5 and subtract it from the rest of the number. Repeat the process, and if the result is divisible by 17, so is the original number.
6. Divisibility by 25: A number is divisible by 25 if its last two digits are divisible by 25 (i.e., if the last two digits are 00, 25, 50, or 75).
7. Divisibility by 125: A number is divisible by 125 if its last three digits are divisible by 125.

These rules allow you to quickly determine divisibility without having to perform long division, saving valuable time during the exam.

Remainder Finding: Using Theorems for Quick Solutions

In the CDS and AFCAT exams, remainder-based problems frequently appear. These problems may ask for the remainder when a large number is divided by another number. To solve these efficiently, several theorems and techniques are useful.

1. Euler’s Theorem:

Euler’s theorem is useful when finding the remainder when a large number is raised to a power and divided by a prime number. While the full theorem involves some advanced concepts, the basic idea is that the remainder of such problems can be determined without calculating the large power directly.

2. Fermat’s Little Theorem:

This theorem is a simplified form of Euler’s Theorem and is used to find remainders when a number is divided by a prime number. If you encounter problems asking for the remainder of large powers divided by a prime, Fermat’s Little Theorem is a handy tool.

Cyclicity of Unit Digits

The unit digit of large powers can also be found using the concept of cyclicity. For instance, the unit digits of powers of a number often follow a repetitive cycle. By identifying this cycle, you can quickly find the unit digit of any large power without performing the entire calculation.

Practical Strategies for Preparing the Number System for CDS and AFCAT

The number system, though fundamental, can be tricky if not approached strategically. Here are some key strategies to help you master the topic:

1. Master Divisibility Rules

• Practice divisibility rules for common numbers like 2, 3, 4, 5, 6, 8, 9, and 10 as these are frequently asked in exams.
• Learn the divisibility rules for larger numbers such as 11, 12, 13, 16, 17, 25, and 125. Practice applying these rules to real numbers to get comfortable with them.
• Create quick reference notes or flashcards for divisibility rules so that you can revise them frequently.

2. Practice Remainder Problems

• Learn how to apply the, Euler’s Theorem, Fermat’s Little Theorem and Other Theorems. These theorems can save a lot of time during exams.
• Use online resources or practice books that contain remainder problems and solutions to get used to applying these theorems in various contexts.
• Focus on understanding the concept of cyclicity when it comes to finding the unit digits of large numbers.

3. Solve Previous Years’ Papers

• Solving previous years’ CDS and AFCAT papers will give you a clear understanding of the type of number system problems that typically appear in these exams.
• Set a timer while solving these papers to simulate exam conditions. Time management is key in competitive exams, and practicing under pressure will help you develop speed.

4. Understand the Basics

• Ensure that your foundational understanding of numbers, including natural, whole, integers, rational, and irrational numbers, is strong. This will help you approach problems logically.
• Focus on prime numbers and their properties, as questions involving divisibility, factors, and multiples often rely on understanding prime numbers.

5. Use Shortcuts and Tricks

• Memorize shortcuts and tricks for solving common number system problems. For example, using simple mental math techniques for divisibility checks can save a lot of time.
• For remainder and unit digit problems, learn the common cyclicity patterns and shortcuts to avoid lengthy calculations.

Conclusion

The Number System is an integral part of the CDS and AFCAT exams, and mastering it requires both conceptual understanding and practice. From divisibility rules to remainder theorems and unit digit problems, the number system tests your ability to solve problems quickly and accurately. By focusing on the key topics discussed in this blog—divisibility rules, remainder finding, and unit digit problems—you can enhance your problem-solving skills and improve your performance in the exams.

Make sure to regularly practice problems from previous years’ papers, utilize shortcuts and tricks, and focus on strengthening your mental math skills. With consistent practice and a strategic approach, mastering the number system will become easier, and you will feel more confident in tackling the quantitative sections of the CDS and AFCAT exams.