## On the sum of divisors

For discussing Olympiad Level Number Theory problems
Masum
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Joined: Tue Dec 07, 2010 1:12 pm

### On the sum of divisors

(Masum Billal): A number has $p$ divisiors where $p$ is a prime.Decide if it has some divisors (excluding itself) such that their sum is equal to the initial number.
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Moon
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### Re: On the sum of divisors

Here goes the hint (hidden below)
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Masum
Posts: 592
Joined: Tue Dec 07, 2010 1:12 pm
Since $p$ prime,ofcourse,$n=q^{p-1}$ for $q$ prime.
First solution:The divisors are $1,q,q^2,...,q^{p-2}$ and there sum is $\frac {q^{p-1}-1} {q-1}$ which is less than $q^{p-1}$.So the sum of some divisors can't be equal to the number.
Second solution:Let $a_0,a_1,...,a_{p-2}$ be a permutation of $0,1,..,p-2$.So if $q^{p-1}=\sum q^{a_i}$ for some $0\le i\le p-2$,then let $a$ be the smallest of $a_i$.And we find contradiction after dividing by $q^a$
One one thing is neutral in the universe, that is $0$.