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Elementary Methods in Number Theory Exercise 1.2.14

Let a,b,c,d be integers such that adbc=1.For integers u and v,define
u=au+bv
v=cu+dv
Prove that (u,v)=(u,v).

 


Proof:
uc=acu+bcv
va=acu+adv
So
ucva=v(bcad)
So
v=vauc
u=dubv
So
(u,v)(u,v)and
(u,v)(u,v)
So
(u,v)=(u,v)


Remark 1:|abcd|=1

(uv)=(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)=(acbd)+(adbc)i

posted @   叶卢庆  阅读(75)  评论(0编辑  收藏  举报
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