### 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 .
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, $$44 = 3 + 41$$.