Elementary Methods in Number Theory Exercise 1.2.14
Let a,b,c,d be integers such that ad−bc=1.For integers u and v,define
u′=au+bv
v′=cu+dv
Prove that (u,v)=(u′,v′).
Proof:
u′c=acu+bcv
v′a=acu+adv
So
u′c−v′a=v(bc−ad)
So
v=v′a−u′c
u=du′−bv′
So
(u,v)≥(u′,v′)and
(u′,v′)≥(u,v)
So
(u,v)=(u′,v′)◻
Remark 1:|abcd|=1(u′v′)=(abcd)(uv)
I think there is some relation to geometric meaning(Liear transformation).But I can't find it at present,maybe it is related to this post.
2.Maybe it is also related to complex numbers.For(a+bi)(−c+di)=(−ac−bd)+(ad−bc)i
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