Question

How many solutions are there to \( 6x + 8y = 218\) such that \(x, \text{ } y \in \mathbb{Z}^+ \)?

Answer 1

An anaylsis shows that:
3x+4y=109
where 109 is odd,
so it must be odd+even,
4y must be even,
so 3x must be odd.
for 3x to be odd, x must be odd.
i.e. list a table for odd values of x, use y= (109-3x)/4 to find y.
This should save some time.