I am looking for some help in solving the equation 15 625x + 8404 = 1024y Thank you
T Trumbone New member Joined Oct 5, 2009 Messages 24 Feb 8, 2010 #1 I am looking for some help in solving the equation 15 625x + 8404 = 1024y Thank you
G galactus Super Moderator Staff member Joined Sep 28, 2005 Messages 7,216 Feb 9, 2010 #2 I will step through this one. Keep it as a guide so you can attempt others. Okey-doke?. 15625x−1024y=−8404\displaystyle 15625x-1024y=-840415625x−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) =−1024(4776)+15625(313)\displaystyle -1024(4776)+15625(313)−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.
I will step through this one. Keep it as a guide so you can attempt others. Okey-doke?. 15625x−1024y=−8404\displaystyle 15625x-1024y=-840415625x−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) =−1024(4776)+15625(313)\displaystyle -1024(4776)+15625(313)−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.
