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

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

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

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    2021-08-13T12:04:20+00:00

    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!

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