N is a perfect number. What is the ratio of the sum of the factor of N and N?
Answers
Answer:
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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)
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Hope this helps!
Given : N is a perfect number.
To Find : The ratio of the sum of the factor of N and N
Solution:
A Perfect Number is a positive integer which is equal to the sum of its Proper divisors/factors.
Perfect number is a number that is sum of all of its positive divisors ( excluding itself )
Perfect number is a number that is half the sum of all of its positive divisors ( including itself )
N is a perfect number
Sum of the factor of N = N + N = 2N
ratio of the sum of the factor of N and N = 2N/N = 2
for example 6 is a perfect number
proper divisors 1 , 2 , 3
Sum = 1 + 2 + 3 = 6 = Number
Factors 1 , 2 , 3 , 6
Sum of Factors = 1 + 2 + 3 + 6 = 12
12/6 = 2
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