Answer
Perfect numbers are defined as the sum of the proper divisors except number should be equal to that positive number and perfect numbers can be determined using Euclid theorem then the perfect number will be equal to \[{{2}^{n-1}}\left( {{2}^{n}}-1 \right)\].
Work Step by Step
Perfect number:
Perfect numbers are defined as the sum of the proper divisors except number should be equal to that positive number and perfect numbers can be determined using Euclid theorem, then the perfect number will be equal to \[{{2}^{n-1}}\left( {{2}^{n}}-1 \right)\].
Here, \[n\]is the prime number and \[n>1\].
Now, substitute the value of n to find perfect numbers.
Then,
If \[n=3\] then, a perfect number is,
\[\begin{align}
& {{2}^{2-1}}\left( {{2}^{2}}-1 \right)=2\times 3 \\
& =6
\end{align}\]
So, the perfect number is \[6\],
And, the divisors of this number are,
\[6=1\times 2\times 3\times 6\]
Now, find the sum of the divisors except the number.
The sum of divisors is \[\left( 1+2+3=6 \right)\].
If \[n=5\] then, a perfect number is,
\[\begin{align}
& {{2}^{5-1}}\left( {{2}^{5}}-1 \right)=16\times 31 \\
& =496
\end{align}\]
So, the perfect number is \[496\].
And, the divisors of this number are:
\[496=1,2,4,8,16,31,62,124,248,496\]
Now, find the sum of the divisors except the number.
The sum of divisors is \[\left( 1+2+4+8+16+31+62+124+248=496 \right)\].
Hence, the above examples are perfect numbers.
