## N is a perfect number. what is the ratio of the sum of the factors of N and N?​

Question

N is a perfect number. what is the ratio of the sum of the factors of N and N?​

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1 month 2021-08-13T12:03:10+00:00 1 Answer 0 views 0

1. Step-by-step explanation:

Consider the number 36.

Canonical Form (expressed in prime numbers): (2^2)*(3^2)

Expressed as factor pairs:

(Note that every factor-pairs together can be expressed in canonical form as follows, because their product is nothing else but the number itself)

Observe carefully!

1*36 = (2^0)*(3^0) * (2^2)*(3^2)

2*18 = (2^1)*(3^0) * (2^1)*(3^2)

3*12 = (2^0)*(3^1) * (2^2)*(3^1)

4*9 = (2^2)*(3^0) * (2^0)*(3^2)

6*6 = (2^1)*(3^1) * (2^1)*(3^1)

Now, our focus is in finding the sum of factors of 36, which is 1+2+3+4+6+9+12+18+36 = 91

This implies:

Sum of factors of 36 =

(2^0)*(3^0) + (2^1)*(3^0) + (2^0)*(3^1) +

(2^2)*(3^0) + (2^1)*(3^1) + (2^0)*(3^2) +

(2^2)*(3^1) + (2^1)*(3^2) + (2^2)*(3^2)

which is equal to

(2^0 + 2^1 + 2^2) * (3^0 + 3^1 + 3^2)

Thus, we can conclude that the sum of factors of a natural number can be expressed based on its canonical form.

i.e, if Canonical form = (2^2)*(3^2), then

Sum of factors =(2^0 + 2^1 + 2^2)*(3^0 + 3^1+3^2)

Mathematics is fun!

Hope this helps!