Find all positive integer solutions for M Find all positive integer solutions M, where $x,y,z$ are non-negative integers from the equations,
$x + y + z = 94$
$4x + 10y + 19z = M$
Attempt:
I multiplied equation 1 by 4, and subtracted it from equation 2 to get
$6y + 15z = M-376$
I know $M-376$ has to be a multiple of 3, and $376 \le M \le 1786$ if we set $(y,x)$ to be $(0,0)$ and $(0,94)$ respectively, but I don't know what to incorporate next. Any clues?
 A: Let $S$ be the set of all combinations of the form $2y+5z$ such that $y$ and $z$ are non-negative integers with $y+z\leq 94$.
You want the set $3S+376$.
So we just need to determine $S$.
We shall prove that $S$ contains every integer in the range $\{1,2,\dots , 94\times 5\}$ except a select few.
Which numbers that are congruent to $1\bmod 5$ can be made? the smallest is clearly $6$ while the largest is clearly $6+ 91\times 5$ ( so we are missing $1,93\times5+1$).
Which numbers that are congruent to $2\bmod 5$ can be made? the smallest is clearly $2$ and the largest is $2+93\times 5$ (so none is missing).
Which numbers that are $3\bmod 5$ can be made? the smallest is clearly $8$ while the largest is $8+90\times 5$ (so we are missing $3,92\times 5+3,93\times 5+3$
Which number that are $4\bmod 5$ can be made? The smallest is clearly $4$ and the largest is $4+92\times 5$ (so we are missing $93\times 5+4$).
Also, it is clear that all of the numbers in between each residue class can also be made, it is also clear that all multiples of $5$ can be made.
Therefore $S=\{1,2,3\dots , 94\times 5\} \setminus \{1,3,92\times 5+3,93\times5+1,93\times 5 +3,93\times 5 + 4\}$
