Satisfying an equation in natural numbers

Mathematics Asked by Luyw on November 26, 2020

I am playing a game where I can gain xp from crafting certain things using some material. The xp is given by$$E=250a+375b+500c$$
$a$ represents how many sets of $50$ of the material I will put in, $b$ of $75$ units, and $c$ of $100$ units. Thus, the total quantity I am putting is
This is divisible by $25$, so I know that if the quantity I have is $Q’=25q+r$, then I will work with $Q=Q’-r$.
It can be seen that $E=5Q$ for any choice of $a$, $b$, and $c$ that satisfies the $Q$ equation.

Given $Q$, how can I find $(a,b,c)$ that satisfy $Q=50a+75b+100c$?

One Answer

Presumably you are asking for solutions in nonnegative integers. As pointed out by Morgan Rodgers in their comment, you can set $Q=25S$ and $(a,b,c)=(25u,25v,25w)$ and solve $S=2u+3v+4w$.

Note that $S$ and $v$ are either both even or both odd. If $S$ is even, then write $S=2T$ and $v=2x$ and solve $T = u+3x+2w$; if $S$ is odd, then write $S=2T+3$ and $v=2x+1$ and solve $T=u+3x+2w$.

In either case, you can choose any values for $x$ and $w$ such that $3x+2y le T$, and then $u=T-3x-2y$ is valid and uniquely determined.

Correct answer by Greg Martin on November 26, 2020

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