Question

Show that 496 is a perfect number.

Answer 1

Hint: Understand the definition of a perfect number. Find all the factors of the given number 496, and add all these factors. If the obtained sum is equal to 496, i.e. the provided number, then it will be a perfect number otherwise not. Do not consider 496 as the factor of itself in the above calculation of sum.

Complete step-by-step solution
Here, we have been given the number 496 and we have to show that it is a perfect number. But first, let us learn about the term ‘perfect number’.
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For example: - let us consider 6, we know that the positive divisors of 6, i.e. the factors of 6 are: - 1, 2, 3, and 6 itself. So, ignoring the number itself, i.e. 6, we get the factors as 1, 2, and 3. Now, taking the sum of these factors, we get,
\[\Rightarrow \] Sum = 1 + 2 + 3
\[\Rightarrow \] Sum = 6
Hence, the sum of the factors of 6, excluding the number itself, is equal to 6. So, 6 is a perfect number.
Now, the sum of divisors of a number, excluding the number itself, is called its aliquot sum, so we can define a perfect number as one that is equal to its aliquot sum.
Now, let us come to the question. Here, we have the number 496. So, So, 496 can be written as: -
\[\begin{align}
  & \Rightarrow 496=1\times 496 \\
 & \Rightarrow 496=2\times 248 \\
 & \Rightarrow 496=4\times 124 \\
 & \Rightarrow 496=8\times 62 \\
 & \Rightarrow 496=16\times 31 \\
\end{align}\]
Therefore, the factors of 496 are: - 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496. So, ignoring the factor itself, i.e. 496, the sum of the other factors can be given as: -
\[\Rightarrow \] Sum = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
\[\Rightarrow \] Sum = 496
Hence, the sum of these factors is equal to the given number. So, 496 is a perfect number.

Note: One must remember the definition of the perfect number to solve the question. Do not get confused in the terms: - perfect number and a perfect square. They have completely different definitions. Now, we can also include the number itself in the sum of factors but then we need to multiply the given number with 2 to get the result. In the above question if we will take the sum of factors including 496 then we will get 992 which is equal to two times the given number 496. So, it will also prove that 496 is a given perfect number. But remember that we generally use the simpler method used in the above solution.