<s>[INST] find all positive integers n with $\sigma(n)=12$ How do you find all positive integers n with $\sigma(n)=12$? 
$\sigma(n)$ is the sum of the divisors of $n$.
The book gives the answer:
Each factor in the formula for $\sigma(n)$ must divide $12$. The only way to get factors, other than $1$, of $12$ for sums of this type are: 
$(1+2)=3$
$(1+3)=4$
$(1+5)=6$
$(1+11)=12$
Hence the only values of $n$ for which $\sigma(n)=12$ are $n=2*3=6$ and $n=11$
I totally do not understand the method, what is the thing I must do to find all positive integers n with $\sigma(n)=12$? And in general $\sigma(n)=k$
 [/INST] Let $n = \prod p_i^{a_i}$ then 
Then the factors are ... any combination of primes and powers.  But the important thing to realize is: Suppose there is a prime factor $p$ and $n$ has $p^a$ power dividing it.  And suppose that $m$ is factor that is not divisible by $p$.  Then $m, pm, p^2m..... p^am$ are also factors.  So them sum of all the factors will be:
$\sigma(n)= (1 + p + p^2 + .... + p^a) + m_1(1+p+p^2 + ..... + p^a) + m_2(1+p+p^2 + ..... + p^a)+ ....$ where $m_1$ are the factors (greater than $1$) that are not divisible $p$.
Do this for all prime factors $p_i$ and you get:
$\sigma(n)= \prod(1 + p_i + .... + p_i^{a_i})$
So $\sigma(n) = 12$ means  $12 = (1 + p + p^2+ ....)(1 + q + q^2 + ...)....$
That is what they mean by "factors in the formula".
But the factors of $12$ are $1,2,3,4,6,12$ so we might have $12 = 12 = (1 + p + p^2 + .....)$.  The only way that can happen is if $p=11$.  So the factors of $n$ are $1$ and $11$. So $n=11$.
So we want to solve any of the following:
a) $1 = (1 + p + p^2 +....)$
b) $2 = (1 + p + p^2 + .....)$
c) $3 = (1 + p + p^2 + .....)$
d) $4 = (1 + p + p^2 + ...)$
e) $6 = (1+ p + p^2 + ...)$ 
f) $12 = (1 + p + p^2 + ..)$.
The solution to a) is $1 = 1$
b) has no solution.
The solution to c) is $3 = 1 + 2$
The solution to d) is $4 = 1+3$
The solution to e) is $6 = 1+ 5$
The solution to f) is $12 = 1 + 11$.
So we have either $12 = (1 + 11)$ and the factors of $n$ are $1,11$ so $n = 11$.
Or we have $12= (1+2)*(1+3)$ and the factors of $n$ are $1,2,3,2*3$. so $n = 6$.
Perhaps abetter example to get the sense of the use of exponent series, would be to find all $n$ so that $\sigma(n) = 56$.
We need $56 = \prod (1 + p_i + ... +p_i^{a_i}$.
The factors of $56$ are $1,2,4,8, 7,14, 28,56$.
$1 = 1$.
$2 \ne 1 + p + p^2 +...$.  Since we can't use $2$ as a factor we can not use $\frac {56}2=28$.
$4 = 1 + 3$
$8 = 1+7$
$7 = 1 + 2 + 4$
$14 = 1 + 13$
$56 \ne 1 + p + p^2$
So we can do either 
$56 = 4*14 = (1 + 3)(1 + 13)$ and $n = 3*13 = 39$
or $56 = 7*8= (1 + 2 + 4)(1+7)$ and $n = 4*7 = 28$.
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