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?
Best Answer
First consider the prime factorization of $1000$. We have:
$$1000=2^3\times 5^3$$
Now, how can we list all the factors of $1000$? We see that we can try listing them in a table:
$$\begin{array}{c|c|c|} & \text{$5^0$} & \text{$5^1$} & \text{$5^2$} & \text{$5^3$} \\ \hline \text{$2^0$} & 1 & 5 & 25 & 125 \\ \hline \text{$2^1$} & 2 & 10 & 50 & 250 \\ \hline \text{$2^2$} & 4 & 20 & 100 & 500 \\ \hline \text{$2^3$} & 8 & 40 & 200 & 1000 \\ \hline \end{array}$$
We see that we can take $(2^1+2^2+2^3) \times (5^0 + 5^1 + 5^2 + 5^3) = 2184$. To get the sum of all factors, we would also include $2^0$ on the left side of the multiplication. We exclude $2^0$ because those would be odd factors.