Show that 496 is a perfect number.

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Detailed Solution

We are given the number 496 and need to show that it is a perfect number.

What is a Perfect Number?

A perfect number is a positive integer that equals the sum of its proper divisors (all positive divisors excluding the number itself). This sum is also called the aliquot sum.

Example (6): Divisors of 6 are 1, 2, 3, 6. Excluding 6: 1 + 2 + 3 = 6, so 6 is perfect.

Divisors of 496:

1, 2, 4, 8, 16, 31, 62, 124, 248, 496

Sum of proper divisors (exclude 496):

1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496

Since the sum equals the number, 496 is a perfect number.

Note

Do not confuse a perfect number with a perfect square—they are different concepts.
If you include the number itself in the sum of divisors for 496: 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 + 496 = 992 = 2 × 496, another confirmation that 496 is perfect.

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