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Perfect numbers are defined as a number for which the sum of all its factors is equal to twice the number itself. It is a positive integer that equals the sum of its divisors except for the number itself.
- Perfect Numbers are the summation of its proper divisors.
- Divisors/factors are those numbers which can divide the given number.
- For e.g.: the factors of 4 are 2,1 because 2 × 2 =4 and 1 × 4 = 4.
- 6 is the smallest perfect number.
- The factor of a number is a number that divides a number exactly without leaving a remainder.
- The factor is always less than or equal to the number.
- During the Greek period, only four perfect numbers were known.
Read More: Remainder Theorem
Key Terms: Perfect number, Divisor, Factorization, Prime Number, Common Factor, HCF, LCM, Integers, Factors, Mersenne prime
What are Perfect Numbers?
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Perfect numbers are those numbers where positive integers is half of the sum of its divisors including the number itself, for eg. The divisors of 6 include 2, 3, 1, 6, the sum of these is 12 and by diving 12 by 2 we get 6 as the answer.
- It is also defined as a positive integer that equals the sum of its divisors except the number itself.
- For example: the divisors of 6 are 2, 3, 1 here 6 is not included in the divisors as per the definition.
- The sum of 2, 3, 1 is 6.
- is prime perfect number.
- is considered to be the even perfect numbers.
- It is based on the Euclid’s Element.
- No exact formula exist for odd perfect numbers.
is prime perfect number.
is considered to be the even perfect numbers.Read More: Multiplication and Division of Integers
Solved Example What are Perfect Numbers?Example: Is 496 a perfect number? Solution: The proper factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248.
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Perfect Number
Read More:
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|---|---|---|
| Irrational Numbers | Real Numbers | Irrational Numbers |
| BODMAS Rule | Fractions | Edges, Faces, and Vertices |
History of Perfect Numbers
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The ancient Greek Mathematician Euclid came up with the idea of perfect numbers. It is speculated that the theory of perfect numbers was first known by the Egyptians. Pythagoras was the first to observe the uniqueness of the number 6, but it was not given much mathematical significance.
- Later Euclid was able to trace a list of such unique numbers.
- He was also able to represent the pattern of these numbers in a formula.
- He concluded the formula for perfect numbers.
N = 2p-1(2p – 1)
Read More: Euclid’s Division Lemma
Mersenne Prime
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Prime numbers are familiar as they are those numbers that do not have other factors other than 1 and the number itself. Mersenne Prime are those prime numbers which are obtained from one less than the power of the prime raised to two. It is given by the formula:
Mp = 2p – 1
- For eg, 2 is a prime number, using the formula Mp = 2p – 1 and substituting 2 in the place of p we get,
- Mp = 2p – 1
- Mp = 22 – 1
- Mp = 4 – 1
- Mp = 3
- They are very closely related with perfect numbers and play a major role in finding them.
The below table help in finding perfect numbers with the use of Mersenne Prime.
| Prime Number | Mersenne Prime Number (2p – 1) |
|---|---|
| 2 | 3 |
| 3 | 7 |
| 5 | 31 |
| 7 | 127 |
| 13 | 8191 |
Read More: Natural Number and Whole Number
How to check if a number is a Perfect Number?
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Steps to check if a number is a perfect number or not are as follows:
- Consider the number and its factors.
- For example, suppose we have a number 50, and its factors are 1, 2, 5, 10, 25, 50.
- As per the perfect numbers definition, the factors of 50 will be 1, 2, 5, 10, and 25 (excluding the number itself).
- Add all the factors which will be represented as 1 + 2 + 5 + 10 + 25 = 43
- If the answer is the same as the number itself, one can conclude that it is perfect.
- If the answer differs, the number is not a perfect number.
- In the given example, we obtained the sum as 43 while the number is 50. Thus, 50 is not a perfect number.
- This method emphasizes executing the definition that can be applied to any number.
- Other methods are devised to check if a number is perfect.
Read More: HCF and LCM
How to find Perfect Numbers
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There are two methods of finding perfect numbers, which are as follows:
Method 1
In this method, first set the number from the beginning. The next number is double the previous number, and such a chain of numbers is continued till the sum of all such numbers is a prime number. The product of the sum of the chain's first and last number of the chain will give a perfect number.
Solved Example of Method 1Example: 1 + 2 + 4 = 7, the chain ended as its sum gave a prime number 7. The product of the sum The last number is 7 × 4 = 28 28 is a perfect number. |
Method 2
Using Mersenne prime (n), \(n(n+1)\over2\) is a perfect number.
Solved Example of Method 2Example: let’s consider the Mersenne prime 3, applying 3 in the formula, Perfect number = \(n(n+1)\over2\) = \(3(3+1)\over2\) = \(3(4)\over 2\) = \(12\over2\) = 6 |
Read More: Value of log 1 to 10
Perfect Numbers Table
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Perfect Numbers Table is tabulated below.
| Prime number, p | Mersenne Prime, 2p – 1 | 2p-1 | Perfect Number 2p-1(2p – 1) |
|---|---|---|---|
| 2 | 3 | 2 | 6 |
| 3 | 7 | 4 | 28 |
| 5 | 31 | 16 | 496 |
| 7 | 127 | 64 | 8128 |
| 13 | 8191 | 4096 | 33550336 |
| 17 | 131071 | 65536 | 8589869056 |
| 19 | 524287 | 262144 | 137438691328 |
| 31 | 2147483647 | 1027371824 | 2305843008139952128 |
Read More: Properties of Real Numbers
Things to Remember
- Perfect numbers are numbers is sum of all the factors excluding the number itself.
- There is no proof that perfect numbers exist beyond the last perfect number found which has 49,724,095 digits.
- All of the perfect numbers are even numbers.
- There are no proof regarding the existence of odd perfect numbers.
- Till date 51 perfect number have been discovered.
- For a number to be a perfect it is necessary that the number must one of the Mersenne prime numbers as its factor.
Read More:
| Related Articles | ||
|---|---|---|
| Is ‘1’ a Prime number? | Addition and Subtraction of Integers | Remainder Theorem |
| Root 2 is an Irrational Number | Exponent Rules | Prime Numbers |
Sample Questions
Ques. What are Perfect Numbers? (2 marks)
Ans. Perfect number is defined as a number for which the sum of all its factors is equal to twice the number itself. It is also defined as a positive integer that equals the sum of its divisors except the number itself. Another definition of perfect numbers is that they are positive integers that are half of the sum of its divisors including the number itself. Eg, 6, 28, 496, 8128.
Ques. Find the factors of the following numbers? (3 marks)
(A) 12
(B)15
(C) 28
(D) 40
Ans. (A)12: 1, 2, 3, 4, 6, 12
(B) 15: 1, 3, 5, 15
(C) 28: 1, 2, 4, 7, 14, 28
(D) 40: 1, 2, 4, 5, 8, 10, 20, 40
Ques. Write the first 5 multiples of the following? (3 marks)
(A) 3
(B) 5
(C) 7
(D) 6
Ans. (A) 3: 3, 6, 9, 12, 15
(B) 5: 5, 10, 15, 20, 25
(C) 7: 7, 14, 21, 28, 35
(D) 6: 6, 12, 18, 24, 30
Ques. Difference between prime number and composite number? (4 marks)
Ans.
| Prime Number | Composite Number |
|---|---|
| Prime number only has two factors | Composite numbers has more than two factors |
| Its factors include 1 and the number itself | Apart from 1 and the number itself, the composite number has many other numbers as its factors. |
| All prime numbers are odd numbers except 2 | Composite numbers have both even and odd numbers except 2. |
| Eg. 2, 3, 7, 11, 13, etc | Eg, 12, 18, 25, 44, etc |
Ques. Which of the following is a perfect number? 13 or 28? (2 marks)
Ans. Solving for 13,
Factors of 13 are 1,13. Excluding 13 the sum will be 1. Thus it does not share similarity thus, it is not a perfect number.
Solving for 28,
Factors of 28 are 1, 2, 4, 7, 14, 28. Excluding 28, we will have the sum of the remaining factors as 28. Since they share similar numbers, 28 is a natural number.
Ques. Find the common factors of 3, 6, 9? (3 marks)
Ans. Factors of 3 – 1, 3
Factors of 6 – 1, 2, 3, 4, 12
Factors of 9 – 1, 3, 9
The common factor is 3.
Ques. Find the common factors of 15 and 25? (2 marks)
Ans. Factors of 15 – 1, 3, 5, 15
Factors of 25 – 1, 5,
The common factor is 5.
Ques. Find the prime factorization of 980? (2 marks)
Ans.
Ques. Find the HCF of 28 and 36. (2 marks)
Ans.
The HCF value of 2 is occurring twice in both 28, 36. Thus the HCF is 2 × 2 = 4.
Ques. Find LCM of 20, 25 and 30? (2 marks)
Ans.
Thus the LCM is 2 × 2 × 3 × 5 × 5
Ques. Find all the perfect numbers from 1 to 400? (3 marks)
Solution: As it is known that every perfect number can be expressed as 2p – 1(2p – 1) where p is a prime number.
- Using the above formula let us find the perfect numbers from 1 to 500.
- For n = 2, 22 – 1(22 – 1) = 2(4 –1) = 2 × 3 = 6.
- For n = 3, 23 – 1(23 – 1) = 22(8 – 1) = 4 × 7 = 28
∴ the perfect numbers between 1 to 400 are 6 and 28
Ques. If we substitute n =10 in 2n−1, will it give us a perfect number? (2 marks)
Ans. Given n = 10
- If 2n−1 is a prime number, then
- 2n−1(2n−1) is a perfect number.
- So, 210−1=1023
- It is not a prime number.
- So, it will not give us a perfect number.
Ques. Find the common factors of 24 and 36? (2 marks)
Ans. Factors of 24 are as follows: 1, 2, 3, 4, 6, 8, 12 and 24
Factors of 36 are as follows: 1, 2, 3, 4, 6, 8, 12 and 24
The common factor is 1, 2, 3, 4, 6, 12.
Ques. Determine the HCF of 18 and 48? (2 marks)
Ans. Factors of 18 are as follows: 1, 2, 3, 6, 9 and 18
Factors of 48 are as follows: 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.
Thus, 6 is the HCF that divides 18 and 48 completely.
Hence, the HCF of 18 and 48 is 6.
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