Perfect Number

Number Theory Level 2

A number is said to be perfect when sum of the positive factors of the number is twice the number.
Example : The positive factors of 6 6 are 1 , 2 , 3 , 6 1,2,3,6
1 + 2 + 3 + 6 = 2 ⋅ 6 1 + 2 + 3 + 6 = 2 \cdot 6

There are many perfect numbers such as 496 , 8128 , 33550336 , 8589869056 496, 8128, 33550336 , 8589869056 etc.
Find the perfect number between 6 6 and 496 496 .


The answer is 28.

Shithil Islam by shithil Islam , 18, Bangladesh

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2 solutions

Margaret Zheng
Feb 29, 2016

Though I think this really depends on pure memorization...... here is one simple explanation.

Here are the factors of 28: 1,2,4,7,14. Adding them, we get 1 + 2 + 4 + 7 + 14 + 28 = 56 1+2+4+7+14+28=56 , and 56 = 2 × 28 56=2 \times 28

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Edwin Gray
Jan 1, 2019

Even perfect numbers are given by the formula 2^(P - 1) (2^P - 1), provided 2^P - 1 is a Prime. (In which case, P must be a prime).Letting P = 3, 2^3 - 1 = 7, a prime. 2^(3 - 1) = 4. So 28 is a perfect number. Ed Gray Interestingly, no one knows whether an odd perfect number exists, although it seems highly unlikely. If one did exist, it must have the form: R^2 (p^a), where p is a prime of the form 4k + 1.

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