Diophantine equation on an example

Question

I do have one task here, that could be solved my guessing the numbers. But the seminars leader said, also Diophantine equation would lead to solution. Has anyone an idea how it works? And could you please show it to me on that example here?

The example:
The number $400$ shall be divided into $2$ summands, the first summand should be divisible through $7$ and the second summand should be divisible through $13$. Thank you for help!

Answer 1

You are saying that $$400 = x + y$$ with $7|x$ and $13|y$.

Or we could just write that as $$400 = 7 v + 13 u.$$

Now notice that $1 = 2 \cdot 7-13$ so $$400 = 7 \cdot 800 - 13\cdot 400$$ gives one solution (of many)

Answer 2

You are looking for solutions to the diophantine equation: $$7x+13y = 400.$$ Clearly, if you can find integer values (x,y) that satisfies this equation, then 7x is your first summand and 13y your second summand.

To solve such a linear diophantine equation, you should use the Euclidean algorithm. Have you done this or do you want me to elaborate a bit more?