A perfect number is a whole number in which the term itself is equal to the sum of all its factors. Given below are some example based on perfect numbers.

Lets consider the number 28
Factors of 28 are 1, 2, 4, 7, 14 and 28
of this the proper factors are 1, 2, 4, 7, 14 and 28
Some of the proper factors = 1+2+4+7+14 = 28
Hence, 28 is a perfect number.
28 is also considered as the only even perfect number.

A perfect number can be defined as an integer, which is a non-zero number. A perfect number can be obtained by adding all the factors which are less than that number. All perfect numbers are even. There is no odd perfect number.

Definition of Perfect Number

It is easy to find unusual properties of small numbers that would characterize these into their peculiarity. The number six (6) has a unique property in which it has both the sum and the product of all its smaller factors; 6 = 1+2+3 or 1 x 2 x 3. By the divisors of a number we mean the factors including unity are less than the number. If the sum of its proper divisors or aliquot divisors is less than the number then we call them as deficient ( as in case of 8). If the total sum of the proper divisors or aliquot divisors exceeds the number as in case of 12, the number is then called abundant. The early Hebrews considered 6 to be a perfect number and Philo Judeus ( 1st century AD) also regarded 6 to be a perfect number.

The Pythagoreans called a number like 6 perfect assuming that the number is the sum of its proper factors which are divisors strictly smaller than the number itself.
The first of the few perfect numbers are as follows: 6, 28, 496, 8128, and 33550336.

Example for Perfect Number

Lets consider the number 6 .
Factors of 6 are 1 , 2 , 3 , and 6
Of this the proper factors are 1 , 2 and 3
Sum of the proper factors = 1+2+3= 6
Hence, 6 is a perfect number.

There are two main types of perfect numbers and they are even perfect numbers which would follow 2n-1 (2n-1)and odd perfect numbers.

Even perfect numbers

Euclid knew that 2n-1 (2n-1) was perfect if 2n-1 is prime.
Euclid proved that, if and when ‘p’ = 1 + 2 + 22 + ….2n is a prime then 2np is referred as a perfect number.
2np is divisible by 1, 2, …2n, p, 2p ..2n-1p is the number less than itself and so the sum of these divisors is 2np.

Odd perfect numbers

This is one of the unresolved problems of number theory.
Eular showed that these numbers have dimensions of pam2, where ‘p’ is prime and p = a = 1 or mod 4.
If ‘n’ is an odd number with s (n) = an, then n < (4d)4K, where ‘d’ is the denominator of a and ‘K’ is the number of distinct prime factors on ‘n’.
If nK is an odd number with k distinct prime factors then n < 44K

Any natural numbers can be expressed as a product of its factors.
When the number is divided by its factors, there will be no reminder.
If the number is excluded from the set of factors of any number, the remaining factors are called proper factors.

If the sum of the proper factors of any number is the number itself, that natural number is called a perfect number.

Example for Proper Factors

Lets consider the number 496 .
Its factors are 1 , 2 , 4 , 8 , 31 , 62 , 124 , 248 and 496 .

Their proper factors are 1 , 2 , 4 , 8 , 31 , 62 , 124 and 248

Sum of the proper factors = 1 + 2 + 4 + 8 + 31 + 62 + 124 + 248 = 496
Hence 496 is a perfect number.

Properties of Perfect Numbers

1. Product of even factors and prime factor gives the number itself:

For a perfect number 6, 2 and 3 are the factors, such that 2 x 3 = 6
As shown above, the product of the even factors (2) and the prime factor (3) gives the number itself (6).
For the perfect number, even factor 2 can be expressed as a power of 2.
2 = 21 and prime factor 3 is can be expressed as
3 = 4 - 1
= 22 - 1

So, 6 = 2 x 3
= 21(22 - 1)
2. Exponent forms of perfect numbers can be expressed as 2(n-1) (2n - 1)
For the perfect number 4960 , its factors 16 multiplied by 31 is 496.
Even factor 16 can be expressed as a power of 2 as 16 = 24
And the prime factor 31 can be written as
31 = 32 - 1
= 25 - 1

So, 496 = 16 x 31
= 24(25 - 1)
So, in general, any even perfect number is of the form 2(n-1) (2n - 1), where n is a prime number.

Example for Properties of Perfect Numbers

Consider the number 2016. Express it as a product of an even number and an odd number. State whether it is a perfect number.

Solution: Factorize 2016.
2016 = 2 x 2 x 2 x 2 x 2 x 3 x 3 x 7

= 25 x 32 x 7

Now consider the factors 25 = 32 and 32 x 7 = 63
So, 2016 = 32 x 63
where 32 is an even number and 63 is an odd number.
This means 2016 can be expressed as 2(n-1)(2n -1) where n=6
So it is a perfect number.

Consider the number 6, number 6 is divisible by 1, 2, 3, and 6 which means after division of 6 by either 1, 2, 3, and 6, we get the remainder as zero. These numbers 1, 2, 3, and 6 are called as factors of 6.

Factors of a number are those numbers which when divided by the given number, leaves a zero remainder.

Examples on Factors of a Number

(i) 3 and 5 are factors of 15 as 3 × 5 = 15

Also, 1 × 15 = 15

or, 1 and 15 are also factors of 15.

Thus, as we discussed above; 1, 3, 5 and 15 are factors of 15 as each of these numbers (1, 3, 5 and 15) divides number 15 exactly.

(ii) 1 and 24 are factors of 24 as 1 × 24 = 24,

$\rightarrow$ 2 and 12 are factors of 24 as 2 × 12 = 24,

$\rightarrow$ 3 and 8 are factors of 24 as 3 × 8 = 24

$\rightarrow$ 4 and 6 are factors of 24 as 4 × 6 = 24

Thus, as discussed above; factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24 as each of these numbers divides number 24 exactly.

(iii) 1 and 48 are factors of 48 as 1 × 48 = 48,

$\rightarrow$ 2 and 24 are factors of 48 as 2 × 24 = 48

$\rightarrow$ 3 and 16 are factors of 48 as 3 × 16 = 48

$\rightarrow$ 4 and 12 are factors of 48 as 4 × 12 = 48

$\rightarrow$ 6 and 8 are factors of 48 as 6 × 8 = 48

Thus, as discussed above; factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48 as each of these numbers divides number 48 exactly.

In other words a factor of a given number divides the given number exactly.

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Let us find those numbers which exactly divide 6. Clearly, 6 is not exactly divisible by any number greater than 6. So, let us divide 6 by any number less than or equal to 6.

We have, 6 ÷ 1 = 6 so, quotient = 6 and remainder = 0.Therefore, 6 = 1 × 6

Again 6 ÷ 2 = 3 so, quotient = 3 and remainder = 0.Therefore, 6 = 2 × 3

Similarly again, 6 ÷ 3 = 2 so, quotient = 2 and remainder = 0.Therefore, 6 = 3 × 2

From this we observe that 1, 2, 3 and 6 are exact divisors of 6.These numbers are called factors of 6.

Thus, we can define the term factor as follows,

A factor of a number is an exact divisor of that number.

In other words, a factor of a number is that number which completely divides the number without leaving a remainder.

Each of the numbers 1, 2, 3, 4, 6 and 12 is a factor of 12. However, none of the numbers 5, 7, 8, 9,10 and 11 is a factor of 12.

For example, 8 divides 40 exactly, therefore 8 is a factor of 40. Similarly, 4 is a factor of 16 and 5 is a factor of 25.

Observe the following:

Multiples

A multiple of a number is a number obtained by multiplying it by a natural number.

If we multiply 3 by 1, 2, 3, 4, 5, 6... ,we get

3 × 1 = 3, 3 × 2 = 6, 3 × 3 = 9, 3 × 4 = 12, 3 × 5 = 15, 3 × 6 = 18 ...

Thus, 3, 6, 9, 12, 15, 18 ... are multiples of 3

Clearly, a number is a multiple of each of its factors.

The multiples of 4 are 4, 8, 12, 16, 20, 24, 28...

Each of these multiples is greater than or equal to 4.

1 is the common factor of every number and every number is always a factor of itself. 1 is the only number which has exactly one factor, so it is a unique number.