The largest coefficient in absolute value in 2x + 3y = 763 is 3 

The nearest multiple of 3 to 763 is 765.

763 = 765 - 2

The LCM of both 2 and 3 is 6.  So we add and subtract a multiple of 6,
namely 6n:

763 = 765 - 2 + 6n - 6n = 6n - 2 + 765 - 6n = 2(3n-1) + 3(255-2n)

Thus 

2(3n-1) + 3(255-2n) = 763  and x = 3n-1 and y = 255-2n

Since y > 0,

255-2n > 0
   -2n > -255
     n < 127.5

So n can range from 1 to 127 inclusively.

Therefore there are 127 solutions in positive integers.

Edwin