/ Preparation Maths Number SystemBy
- Introduction
- What are the Perfect Numbers?
- Euclid Euler Theorem
- Illustrative Examples
- Frequently Asked Questions
Introduction
The world of mathematics is full of concepts with no direct use in nature or science, but are fascinating nevertheless. Numbers can create all sorts of exciting patterns and develop new concepts, definitions, and techniques. One such concept is the topic of perfect numbers.
What are the Perfect Numbers?
A perfect number in mathematics is a number that is numerically equal to the additive of its divisors apart from itself. These divisors include both prime factors and non-prime factors, all taken just once. For example, the number 6 has factors 1, 2, 3, and 6. Now, the sum of all its factors except itself is 1 + 3 + 2 = 6. Hence, 6 may be called a perfect number. The sum of all the divisors of a number except itself is known as the number’s aliquot sum. Hence, a perfect number can be said to be a number that is equal to its aliquot sum.
Another example of a perfect number is 28. The number 28 may be expressed in the following ways: 1 x 28, 2 x 14, 4 x 7, 7 x 4, 14 x 2, and 28 x 1. Hence, the divisors of 28, except itself, are 1, 2, 4, 7, and 14. The aliquot sum of the number 28 is 28, making it a perfect number.
Euclid Euler Theorem
Euclid’s Elements is the world’s earliest complete treatise on mathematics. The concept of perfect numbers is also mentioned in Euclid’s Elements, referred to as a complete number. Euclid was also able to prove in his treatise a rule for even perfect numbers. For this rule, consider q to be a prime number of the form 2p - 1. This is called a Mersenne prime. Now, the number q(q+1)/2 is a perfect number. Euler was later able to prove that all even numbers have this in common. This theorem is hence called the Euclid Euler Theorem.
About the Topic
In the NCERT syllabus, perfect numbers is a rather short topic generally reserved only for short answers or multiple-choice questions. They are not used for medium or long answer questions. Generally, this topic will not appear for more than 1 or 2 marks in an exam.
Illustrative Examples
1. Prove that 28 is a perfect number.
The divisors of 28, except itself, are 1, 2, 4, 7, and 14. The aliquot number 28 is 28, making it a perfect number.
2. Is 20 a perfect number?
The divisors of 20, except itself, are 1, 2, 4, 5, and 10. The aliquot sum of the number 20 is 22, making it an imperfect number.
3. Is 6 a perfect number?
For p = 2, 2p - 1 = 3. 3(3+1)/2 = 6. Hence 6 is a perfect number.
Frequently Asked Questions
Q: What are the first few perfect numbers?
Q: How many odd perfect numbers are known?
Q: Which perfect number is the sum of two cubes?
Q: What is an aliquot sum?
Q: How can a perfect number be defined in terms of aliquot sum?
Maths Number System Exam
Student Forum
Anything you would want to ask experts?Answered 3 weeks ago
Candidates should first understand basic concepts, including divisibility rules, prime factorisation, HCF and LCM, and remainders. They should then solve topic-wise practice questions and previous years' CAT questions to improve accuracy and problem-solving speed.
S
Contributor-Level 7
Answered 3 weeks ago
There is no fixed number of questions from the Number System in CAT. The number of questions varies across exam slots and years. Candidates should prepare the topic along with other important areas of Quantitative Aptitude.
A
Contributor-Level 7
Answered 3 weeks ago
Yes, the Number System is an important topic for CAT Quantitative Aptitude. Questions may test concepts such as divisibility, remainders, factors, and properties of integers. Candidates should also practise applying these concepts to different question types.
J
Contributor-Level 7
Answered 3 weeks ago
The CAT Number System covers topics such as divisibility rules, factors and multiples, prime numbers, HCF and LCM, remainders, unit digits, factorials, integers, rational numbers, and properties of numbers.
Contributor-Level 10
Answered 2 months ago
CLAT does not publish a fixed count for Number System questions. The UG exam stays largely passage-based. This topic supports calculation inside Quantitative Techniques passages. It does not appear as standalone questions.
S
Contributor-Level 10
Answered 2 months ago
Number System questions cover many types. These include basic property checks, combined divisibility conditions, HCF and LCM problems, remainder patterns and multi condition reasoning. They test calculation speed and accuracy.
S
Contributor-Level 10
Answered 2 months ago
Students can improve step by step. Learn divisibility rules. Practise HCF and LCM daily. Work through remainder questions carefully. Regular revision of common traps also builds accuracy and speed.
S
Contributor-Level 10
Answered 2 months ago
There is no single fixed book for this topic. Students often use standard Class 10 level maths books. Combine these with regular practice sets and previous year papers to build accuracy.
S
Contributor-Level 10
Answered 2 months ago
Students should start early. Practice Number System alongside other Quantitative Techniques topics. Strong basics make combined divisibility and multi condition questions easier later.
S
Contributor-Level 10
Answered 2 months ago
Number System is not a separate scored CLAT topic. But the calculation skills help with passage-based Quantitative Techniques questions. Skipping it fully is not wise. Most passages need quick, accurate number work.
S
Contributor-Level 10
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