I will step through this one. Keep it as a guide so you can attempt others. Okey-doke?.
\(\displaystyle 15625x-1024y=-8404\)
Find GCD(15625, -1024):
-1024=(-1)15625+14601
15625=14601(1)+1024
14601=1024(14)+265
1024=265(3)+229
265=229(1)+36
229=36(6)+13
36=13(2)+10
13=10(1)+3
3=3(1)+0
GCD=1
Now, back substitute to find the linear combination so we can express the GCD in terms of -1024 and 15625:
1=(1*10)+(-3*3)
=(-3*13)+(4*10)
=(4*36)+(-11*13)
=(-11*229)+(70*36)
=(70*265)+(-81*229)
=(-81*1024)+(313*265)
=(313*14601)+(-4463*1024)
=(-4463*15625)+(4776*14601)
=\(\displaystyle -1024(4776)+15625(313)\)
Therefore, a particular solution is
x=-2630452 and y=-40137504
The general solution works out to be
x=-2630452+1024t and y=-40137504+15625t
It is tedious, but is pretty cool once you get the hang of it.
