Statement Find two integers xxx and yyy such that 84x+60y=gcd(84,60).84x + 60y = \gcd(84,60).84x+60y=gcd(84,60). Solution By the Euclidean algorithm: 84=1⋅60+24, 60=2⋅24+12, 24=2⋅12+0,\begin{aligned} 84 &= 1\cdot 60 + 24, \ 60 &= 2\cdot 24 + 12, \ 24 &= 2\cdot 12 + 0, \end{aligned}84=1⋅60+24, 60=2⋅24+12, 24=2⋅12+0, so gcd(84,60)=12\gcd(84,60)=12gcd(84,60)=12. Back-substituting: 12=60−2⋅24=60−2 (84−60)=3⋅60−2⋅84.12 = 60 - 2\cdot 24 = 60 - 2\,(84 - 60) = 3\cdot 60 - 2\cdot 84.12=60−2⋅24=60−2(84−60)=3⋅60−2⋅84. Hence one solution is x=−2, y=3,84(−2)+60(3)=−168+180=12.\boxed{x=-2,\ y=3},\qquad 84(-2)+60(3) = -168+180 = 12.x=−2, y=3,84(−2)+60(3)=−168+180=12.