Fombitz thought you meant all solutions, but you probably meant
only solutions (x,y) where x and y are both whole numbers:
3x + 7y = 999
Write all the numbers in terms of the nearest multiple to the absolute
value of the coefficient of the variable with the smallest absolute
value that are not already a multiple of it:
Of the numbers 3, 7, 333, only 7 is not a multiple of 3, so we write
7 in terms of its nearest multiple of 3, which is 6. So we write
7 as 6+1
3x+(6+1)y = 999
3x+6y+y = 999
WE divide every term by 3
x+2y+y/3 = 333
y/3 = 333-x
Since 333-x is a non-negative integer, so is y/3.
Let the integer be A, then
333-x = A and y/3 = A
So y = 3A
We substitute that in
3x + 7y = 999
3x + 7(3A) = 999
3x + 21A = 999
Divide through by 3
x + 7A = 333
x = 333-7A
Now since x >= 0,
333-7A >= 0
-7A >= -333
A <= 333/7 = 47 4/7
A <= 47
Also since y >= 0
3A >= 0
A >= 0
So there are 48 solutions from A = 0 through 47 inclusively.
Edwin
