I have tried to solve this problem:
714x + 7007y = 7
but I have not come to a solution yet.
Thank you!
Let us find the greatest common divisor by the algorithm mentioned in the Elements.
7007=(9)714+581
714=(1)581+133
581=(4)133+49
133=(2)49+35
49=(1)35+14
35=(2)14+7
14=(2)7+0
Work backwards.
7=35-2*14
7=35-2(49-35)=3*35-2*49
7=3(133-2*49)-2*49
7=3*133-8*49
7=3*133-8(581-4*133)=35*133-8*581
7=35(714-581)-8*581=35*714-43*581
7=35*714-43(7007-9*714)=422*714-43*7007
Thus,
714x+7007y=7
Say for example,
x=422 and y=-43