Question 2

2. Find all factors of 496 and prove that 496 is a perfect number.

All the factors of 496 are:
1, 2, 4, 8, 16, 31, 62, 124, 248, and 496.
How do I know that these are ALL of the factors of 496? I used a factoring tree!

Using the factoring tree, I can determine all of the possible factors of 496.
I know:
2 x 248 = 496
2 x 2 x 124 = 496; 4 x 124 = 496
2 x 2 x 2 x 62 = 496; 8 x 62 = 496
2 x 2 x 2 x 2 x 31 = 496; 16 x 31 = 496
How do I know that two is a factor of 496 and how do I know to stop at 31? I know that as long as a number is even, it can be factored over and over using the number two. I also know that 31 is a prime number using the last problem and because it is no longer factorable, these are all the possible factors of 496.

496 is a perfect number. How do I know this? First, I have to understand what a perfect number is. A perfect number is whole, positive integer that is equal to the sum of all of its factors, excluding itself.
                For example, 6 is a perfect number.
                I know that the factors of 6 are: 1, 2, 3, and 6.
                1 + 2 + 3 = 6
Then, what about 496?
1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496
(1 + 2 + 4 + 8 +16  = 31)
31 + 31 + 62 + 124 + 248 = 496
(31 + 31 = 62)
62 + 62 + 124 + 248 =  496
(62 + 62 = 124)
124 + 124 + 248 = 496
(124 + 124 = 248)
248 + 248 = 496

Bonus - Do the same for 8.128 and 8,589,869,056.
Using the same method, I found all of the factors of 8,128 are:
1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064 and 8128
So, is it a perfect number? Let's add all of the factors except for itself together:
1+2+4+8+16+32+64+127+254+508+1016+2032+4064 = 8128.
This works! Therefore, 8128 is a perfect number.
What about 8,589,869,056? What are all of its factors? The factors of 8,589,869,056 are:
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131071, 262142, 524284, 1048568, 2097136, 4194272, 8388544, 16777088, 33554176, 67108352, 134216704, 168433408, 536866816, 1073733632, 2147467264, 4294934528, 8589869056
So, is it a perfect number?
Let's add!
1+2+4+8+16+32+64+128+256+512+1024+2048+4096+8192+16384+32768+65536+131071+262142
+524284+1048568+2097136+4194272+8388544+16777088+33554176+67108352+134216704+ 168433408+536866816+1073733632+2147467264+4294934528 = 8,589,869,056
This is true so it means that 8,589,869,056 is a perfect number!