TITLE: What is the sum of all positive even divisors of 1000?
QUESTION [24 upvotes]: I know similar questions and answers have been posted here, but I don't understand the answers. Can anyone show me how to solve this problem in a simple way? This is a math problem for 8th grade students.Thank you very much!

What is the sum of all positive even divisors of 1000?

REPLY [2 votes]: We want to exclude the odd divisors, and the odd divisors of $1000$ are exactly the divisors of $125$. Therefore your answer is the sum of divisors of $1000$, minus the sum of divisors of $125$, or using $\sigma$ for the sum of divisors,
$$
\sigma(1000) - \sigma(125).
$$
Now, to compute $\sigma(n)$, you split $n$ into its relatively prime parts and then sum the divisors of each part. So we get
\begin{align*}
\sigma(1000) - \sigma(125)
&= \sigma(125) \sigma(8) - \sigma(125) \\
&= (1 + 5 + 25 + 125)(1 + 2 + 4 + 8) - (1 + 5 + 25 + 125) \\
&= 156 \cdot15 - 156 \\
&= 156 \cdot 14 \\
&= 2184.
\end{align*}