Number System is an important topic in mathematics and quantitative aptitude. It is useful for school exams, government job exams, banking exams, SSC, railway exams, entrance tests and placement tests. Many questions from this topic look short, but they need a clear understanding of formula, method and calculation steps.
This lesson explains number system in simple language. It includes meaning, important terms, formulas, shortcut ideas, solved examples, common mistakes, practice questions and frequently asked questions. The aim is to make the topic easy to learn for beginners and useful for exam preparation.
Before solving any question, read the statement carefully and write the given values separately. Then decide which formula or method is required. This small habit prevents most calculation mistakes.
What Is Number System?
Number system is the study of numbers, their types, properties and operations. It includes natural numbers, whole numbers, integers, rational numbers, irrational numbers, prime numbers, composite numbers, factors and multiples.
In aptitude exams, number system questions usually test basic understanding, speed of calculation and the ability to choose the correct method quickly. A good answer should be accurate, but it should also be solved in less time.
Important Terms
| Point | Details |
|---|---|
| Natural numbers | Counting numbers 1, 2, 3 and so on |
| Whole numbers | Natural numbers including 0 |
| Integers | Positive, negative and zero numbers |
| Prime numbers | Numbers with exactly two factors |
| Composite numbers | Numbers with more than two factors |
Key Formulas
| Point | Details |
|---|---|
| Even number | A number divisible by 2 |
| Odd number | A number not divisible by 2 |
| Prime test | Check divisibility up to square root |
| Dividend | Divisor x Quotient + Remainder |
| Divisibility by 9 | Sum of digits divisible by 9 |
How to Solve Number System Questions
The best method is to move step by step. First identify the given data. Second, decide what is asked. Third, choose the formula. Fourth, substitute values carefully. Fifth, check whether the answer has the correct unit or form.
For multiple-choice aptitude questions, also look at the options after doing the basic setup. Sometimes the options help you avoid long calculations. Still, never guess before understanding the question because many options are designed around common mistakes.
- Identify the type of number in the question.
- Use divisibility rules before long division.
- For remainders, use dividend = divisor x quotient + remainder.
- For prime checking, test factors up to square root.
- For factors and multiples, separate HCF and LCM type language.
Shortcut Ideas
- A number divisible by 2 and 3 is divisible by 6.
- A number is divisible by 4 if last two digits are divisible by 4.
- A number is divisible by 8 if last three digits are divisible by 8.
- For 9, add digits and check divisibility by 9.
- For 11, compare alternate digit sums.
Common Question Types
| Point | Details |
|---|---|
| Number classification | Identify type of number |
| Divisibility | Check if number is divisible |
| Remainder | Find remainder after division |
| Unit digit | Find last digit of powers |
| Factors | Count or find factors |
Solved Examples
Example 1
Is 4374 divisible by 9?
Digit sum = 4 + 3 + 7 + 4 = 18. Since 18 is divisible by 9, 4374 is divisible by 9.
Example 2
Find remainder when 57 is divided by 8.
8 x 7 = 56, remainder = 1.
Example 3
Is 29 prime?
Square root of 29 is between 5 and 6. Check 2, 3, 5. None divides 29. So 29 is prime.
Example 4
Find the smallest 3-digit number divisible by 12.
Smallest 3-digit number is 100. Next multiple of 12 is 108.
Example 5
Find unit digit of 7 squared.
7 x 7 = 49, so unit digit is 9.
Exam Style Examples with Explanation
Exam Example 1
Find the greatest 4-digit number divisible by 15.
Greatest 4-digit number is 9999. 15 x 666 = 9990. Required number is 9990.
For greatest/smallest divisible number, use nearest multiple.
Exam Example 2
A number leaves remainder 3 when divided by 7. What remainder will twice the number leave?
Twice the remainder = 6, so remainder is 6.
Remainders can be multiplied and then reduced by divisor.
Exam Example 3
Check if 12321 is divisible by 11.
Alternate sums: 1 + 3 + 1 = 5 and 2 + 2 = 4. Difference = 1, so not divisible by 11.
For 11, alternate sum difference must be 0 or multiple of 11.
Common Mistakes
- Calling 1 a prime number.
- Forgetting that 0 is a whole number.
- Using divisibility by 3 rule for 9 without checking 9.
- Testing too many factors for prime numbers.
- Confusing factor with multiple.
Practice Questions
- Is 246 divisible by 6?
- Is 784 divisible by 8?
- Find remainder when 100 is divided by 9.
- Is 37 prime?
- Find smallest 4-digit number divisible by 25.
- Find greatest 3-digit number divisible by 11.
- Check divisibility of 729 by 9.
- Find unit digit of 6^25.
- Is 91 prime?
- Write first five multiples of 12.
Answers to Practice Questions
- Yes
- Yes
- 1
- Yes
- 1000
- 990
- Yes
- 6
- No, 91 = 7 x 13
- 12, 24, 36, 48, 60
Frequently Asked Questions
What is number system?
It is the study of numbers and their types, properties and operations.
Is 1 prime?
No, 1 is neither prime nor composite.
What is an integer?
An integer is a positive number, negative number, or zero without decimal part.
What is a factor?
A factor divides a number exactly.
What is a multiple?
A multiple is obtained by multiplying a number by whole numbers.
Revision Notes
Number system questions become easy when number types and divisibility rules are clear.
For fast revision, remember the formula, one easy example and one common mistake. If you can explain the topic to another student in simple words, your concept is strong enough for most exam questions.
More Practice for Accuracy
Practice is very important in number system. Start with direct formula questions, then solve word problems, and finally try mixed questions. While practising, write down every mistake in a small notebook. After a few days, revise only those mistakes. This improves speed and accuracy together.
In government exam questions, the language may be slightly different, but the concept remains the same. Look for keywords such as total, difference, ratio, remaining, at least, greatest, smallest, average, possible ways, or probability. These words tell you which formula should be used.
Do not try to memorise every solved question. Instead, understand the pattern behind it. Once the pattern is clear, you can solve many similar questions even when the numbers are changed.
Conclusion
Number System is the base of arithmetic, so strong basics here help in almost every maths topic.
Learn the basic rule first, practise solved examples, and then attempt practice questions without looking at the answers. This is the best way to become confident in number system.
Additional Exam Note 11
Number System questions become easier when you slow down for the first few seconds and understand the condition. Many wrong answers happen because students directly apply a remembered formula without checking whether the question asks for value, ratio, number of ways, time, quantity, or final answer.
While solving, keep units consistent and simplify numbers whenever possible. If the question contains fractions, convert them carefully. If it contains percentages, write them as fractions when that makes the calculation faster. If it contains a ratio, use variables such as x, 2x, 3x or 5x to keep the relation clear.
A useful practice method is to solve the same example in two ways: one by formula and one by logical reasoning. This builds confidence and helps in exams where time is limited. After solving, check whether the answer is reasonable. For example, an average should lie between the smallest and largest value, and a probability should never be greater than 1.
Additional Exam Note 12
Number System questions become easier when you slow down for the first few seconds and understand the condition. Many wrong answers happen because students directly apply a remembered formula without checking whether the question asks for value, ratio, number of ways, time, quantity, or final answer.
While solving, keep units consistent and simplify numbers whenever possible. If the question contains fractions, convert them carefully. If it contains percentages, write them as fractions when that makes the calculation faster. If it contains a ratio, use variables such as x, 2x, 3x or 5x to keep the relation clear.
A useful practice method is to solve the same example in two ways: one by formula and one by logical reasoning. This builds confidence and helps in exams where time is limited. After solving, check whether the answer is reasonable. For example, an average should lie between the smallest and largest value, and a probability should never be greater than 1.
Additional Exam Note 14
Number System questions become easier when you slow down for the first few seconds and understand the condition. Many wrong answers happen because students directly apply a remembered formula without checking whether the question asks for value, ratio, number of ways, time, quantity, or final answer.
While solving, keep units consistent and simplify numbers whenever possible. If the question contains fractions, convert them carefully. If it contains percentages, write them as fractions when that makes the calculation faster. If it contains a ratio, use variables such as x, 2x, 3x or 5x to keep the relation clear.
A useful practice method is to solve the same example in two ways: one by formula and one by logical reasoning. This builds confidence and helps in exams where time is limited. After solving, check whether the answer is reasonable. For example, an average should lie between the smallest and largest value, and a probability should never be greater than 1.
Additional Exam Note 16
Number System questions become easier when you slow down for the first few seconds and understand the condition. Many wrong answers happen because students directly apply a remembered formula without checking whether the question asks for value, ratio, number of ways, time, quantity, or final answer.
While solving, keep units consistent and simplify numbers whenever possible. If the question contains fractions, convert them carefully. If it contains percentages, write them as fractions when that makes the calculation faster. If it contains a ratio, use variables such as x, 2x, 3x or 5x to keep the relation clear.
A useful practice method is to solve the same example in two ways: one by formula and one by logical reasoning. This builds confidence and helps in exams where time is limited. After solving, check whether the answer is reasonable. For example, an average should lie between the smallest and largest value, and a probability should never be greater than 1.
Additional Exam Note 17
Number System questions become easier when you slow down for the first few seconds and understand the condition. Many wrong answers happen because students directly apply a remembered formula without checking whether the question asks for value, ratio, number of ways, time, quantity, or final answer.
While solving, keep units consistent and simplify numbers whenever possible. If the question contains fractions, convert them carefully. If it contains percentages, write them as fractions when that makes the calculation faster. If it contains a ratio, use variables such as x, 2x, 3x or 5x to keep the relation clear.
A useful practice method is to solve the same example in two ways: one by formula and one by logical reasoning. This builds confidence and helps in exams where time is limited. After solving, check whether the answer is reasonable. For example, an average should lie between the smallest and largest value, and a probability should never be greater than 1.
Additional Exam Note 19
Number System questions become easier when you slow down for the first few seconds and understand the condition. Many wrong answers happen because students directly apply a remembered formula without checking whether the question asks for value, ratio, number of ways, time, quantity, or final answer.
While solving, keep units consistent and simplify numbers whenever possible. If the question contains fractions, convert them carefully. If it contains percentages, write them as fractions when that makes the calculation faster. If it contains a ratio, use variables such as x, 2x, 3x or 5x to keep the relation clear.
A useful practice method is to solve the same example in two ways: one by formula and one by logical reasoning. This builds confidence and helps in exams where time is limited. After solving, check whether the answer is reasonable. For example, an average should lie between the smallest and largest value, and a probability should never be greater than 1.
Additional Exam Note 20
Number System questions become easier when you slow down for the first few seconds and understand the condition. Many wrong answers happen because students directly apply a remembered formula without checking whether the question asks for value, ratio, number of ways, time, quantity, or final answer.
While solving, keep units consistent and simplify numbers whenever possible. If the question contains fractions, convert them carefully. If it contains percentages, write them as fractions when that makes the calculation faster. If it contains a ratio, use variables such as x, 2x, 3x or 5x to keep the relation clear.
A useful practice method is to solve the same example in two ways: one by formula and one by logical reasoning. This builds confidence and helps in exams where time is limited. After solving, check whether the answer is reasonable. For example, an average should lie between the smallest and largest value, and a probability should never be greater than 1.
Additional Exam Note 22
Number System questions become easier when you slow down for the first few seconds and understand the condition. Many wrong answers happen because students directly apply a remembered formula without checking whether the question asks for value, ratio, number of ways, time, quantity, or final answer.
While solving, keep units consistent and simplify numbers whenever possible. If the question contains fractions, convert them carefully. If it contains percentages, write them as fractions when that makes the calculation faster. If it contains a ratio, use variables such as x, 2x, 3x or 5x to keep the relation clear.
A useful practice method is to solve the same example in two ways: one by formula and one by logical reasoning. This builds confidence and helps in exams where time is limited. After solving, check whether the answer is reasonable. For example, an average should lie between the smallest and largest value, and a probability should never be greater than 1.
Additional Exam Note 24
Number System questions become easier when you slow down for the first few seconds and understand the condition. Many wrong answers happen because students directly apply a remembered formula without checking whether the question asks for value, ratio, number of ways, time, quantity, or final answer.
While solving, keep units consistent and simplify numbers whenever possible. If the question contains fractions, convert them carefully. If it contains percentages, write them as fractions when that makes the calculation faster. If it contains a ratio, use variables such as x, 2x, 3x or 5x to keep the relation clear.
A useful practice method is to solve the same example in two ways: one by formula and one by logical reasoning. This builds confidence and helps in exams where time is limited. After solving, check whether the answer is reasonable. For example, an average should lie between the smallest and largest value, and a probability should never be greater than 1.
Additional Exam Note 25
Number System questions become easier when you slow down for the first few seconds and understand the condition. Many wrong answers happen because students directly apply a remembered formula without checking whether the question asks for value, ratio, number of ways, time, quantity, or final answer.
While solving, keep units consistent and simplify numbers whenever possible. If the question contains fractions, convert them carefully. If it contains percentages, write them as fractions when that makes the calculation faster. If it contains a ratio, use variables such as x, 2x, 3x or 5x to keep the relation clear.
A useful practice method is to solve the same example in two ways: one by formula and one by logical reasoning. This builds confidence and helps in exams where time is limited. After solving, check whether the answer is reasonable. For example, an average should lie between the smallest and largest value, and a probability should never be greater than 1.
Additional Exam Note 27
Number System questions become easier when you slow down for the first few seconds and understand the condition. Many wrong answers happen because students directly apply a remembered formula without checking whether the question asks for value, ratio, number of ways, time, quantity, or final answer.
While solving, keep units consistent and simplify numbers whenever possible. If the question contains fractions, convert them carefully. If it contains percentages, write them as fractions when that makes the calculation faster. If it contains a ratio, use variables such as x, 2x, 3x or 5x to keep the relation clear.
A useful practice method is to solve the same example in two ways: one by formula and one by logical reasoning. This builds confidence and helps in exams where time is limited. After solving, check whether the answer is reasonable. For example, an average should lie between the smallest and largest value, and a probability should never be greater than 1.
Additional Exam Note 29
Number System questions become easier when you slow down for the first few seconds and understand the condition. Many wrong answers happen because students directly apply a remembered formula without checking whether the question asks for value, ratio, number of ways, time, quantity, or final answer.
While solving, keep units consistent and simplify numbers whenever possible. If the question contains fractions, convert them carefully. If it contains percentages, write them as fractions when that makes the calculation faster. If it contains a ratio, use variables such as x, 2x, 3x or 5x to keep the relation clear.
A useful practice method is to solve the same example in two ways: one by formula and one by logical reasoning. This builds confidence and helps in exams where time is limited. After solving, check whether the answer is reasonable. For example, an average should lie between the smallest and largest value, and a probability should never be greater than 1.