Theory:

A perfect number is defined as a number for which the sum of all its factors is equal to twice the number.
Example:
Let's take the factors of \(6\) are \(1\), \(2\), \(3\) and \(6\).
 
Then the sum of the factors becomes:
 
\(1+2+3+6=12 =12 \times6\)
 
Therefore, \(6\) is a perfect number.
 
Likewise the factors of \(6\), we take the factors of \(496\) are \(1\), \(2\), \(4\), \(8\), \(16\), \(31\), \(62\), \(124\), \(248\) and \(496\).
 
\(1+2+4+8+16+31+62+124+248+496\) \(= 992\) \(=2\times 496\)
 
Therefore, \(496\) is a perfect number.
A co-prime number is a number with any set of numbers that do not have any other common factor other than \(1\). It is also known as relatively prime numbers.
Example:
Factors of \(5 =\) \(1\), \(5\).
 
Factors of \(6 =\) \(1\), \(2\), \(3\), \(6\).

These two numbers show that \(5\) and \(6\) have no common factor other than 1.

Therefore, they are co-prime numbers.
Properties of co-prime numbers:
  • All prime numbers are co-prime to each other.
  • Any consecutive whole numbers are always co-primed.
  • Sum of any two co-prime numbers is always co-primed.
  • Co-prime numbers need not be prime numbers.
Twin primes are a pair of primes that differ by \(2\).
First few twin primes are (3, 5); (5, 7); (11, 13); (17, 19); (29, 31); (41, 43).
Example:
Express \(44\) as the sum of two odd primes.
 
Here we have to find \(2\) numbers which are odd as well as prime numbers.
 
And whose sum is \(44\).
 
Odd prime numbers upto \(44\) are \(3\), \(5\), \(7\), \(11\), \(13\), \(17\), \(19\), \(23\), \(29\), \(31\), \(37\), \(41\), \(43\).
 
Now let’s find out a pair of numbers whose sum is \(44\). Sum of \(3\) and \(41\) is \(44\).
 
So, \(44 = 3 + 41\).