The number of positive integral solution of 2x + 3y = 763 is

1) 125

2) 126

3) 127

4) 128

Plz answer with explanation !!

Question

The number of positive integral solution of 2x + 3y = 763 is
1) 125
2) 126
3) 127
4) 128
Plz answer with explanation !!

Ayush Jha · Accepted Answer

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
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