Find x and y such that 27x 15y 27 15 with x positive but a

Find x and y such that 27x + 15y = (27, 15) with x positive but as small as possible.

Solution

(27,15)=3

-1*27+15*2=3

General solution is

-(1+15k)*27+15*(2+27k)=3

k is any integer

Setting k=-1 gives the smallest positive value of x

x=-(1+15*(-1))=14, y=2-27=-25

x=14,y=-25