公式可变形为y=14x/(x-21),因为y为整数,所以14/(x-21)为整数,或x/(x-21)为整数
①若14/(x-21)为整数,则x-21={-14,-7,-2,-1,1,2,7,14}
对应x={7,14, 19, 20,22,23,28,35}
对应y={-7,-28,-133,-280,308,161,56,35}
②若x/(x-21)为整数m,即x/(x-21)=m,变形得x=21m/(m-1),m/(m-1)为整数或21/(m-1)为整数。
对于整数m,除m=2和m=0外,m/(m-1)均非整数;当m=2时,x=42, y=28;当m=0时,x=y=0不符合题意,舍去。
若21/(m-1)为整数,则m-1={-21, -7,-3,-1, 1,3,7,21}
对应m={-20,-6,-2,0,2,4,8,22}
对应x={-1,-3,-7,-21,21,7,3,1}
对应y={0.636363636,1.75,3.5,7,#DIV/0!,-7,-2.333333333,-0.7}
其中x,y均为整数的解为x=-21,y=7和,x=7和y=-7
综上,全部整数解为:
x=7, y=-7
x=14, y=-28
x=19, y=-133
x=20, y=-280
x=22, y=308
x=23, y=161
x=28, y=56
x=35, y=35
x=42, y=28
x=-21, y=7
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