Dividing foods, does anybody know what kind of problem this is?

Question

There are foods given to participants of a competition. As listed below :

Every participants will get a box of food A

Every 3 participants will get a box of food B

Every 5 participants will get a box of food C

Every 7 participants will get a box of food D

Every 9 participants will get a box of food E

It is known that there are a total of 3378 boxes of food. How many participants are there in the competition?

I can do this by testing the answers 1 by 1, but sometimes there are similar answer. Does anybody know how to solve this? Or the method? And also whats the subject name. Thanks for anyone who is willing to help :)

Oh yeah, I got 1890 because when I divide it by 1, 3, 5, 7, 9 it has no remainder

Answer 1

We have $$n+\frac n3+\frac n5+\frac n7+\frac n9=3378\ .$$ Multiply by $9$ and $7$ and $5$: $$315n+105n+63n+45n+35n=3378\times5\times7\times9\ .$$ Collect terms: $$563n=1064070\ .$$ Divide: $$n=1890\ .$$

Answer 2

Consider a simpler problem in which every participant gets a box $A$ and every 2 participants get a box $B$. There are $102$ boxes. Let $n$ be the number of participants. Then, $$ n+\frac{n}{2}=102. $$ Multiplying this equation by $2$, we get $$ 2n+n=204. $$ Adding up the common factors, we get $$ 3n=204. $$ Dividing both sides by $3$, we get $n=68$. Can you generalize the approach to your problem?