Monday, 21 July 2014

number theory - Euclidean algorithm on (an1,am1)



I'm having trouble applying the Euclidean algorithm to
gcd(an1,am1)=agcd(n,m)1


I've seen others do it such as in this solution but I'm having trouble figuring out what they are doing in each step.



Could someone help explain this to me?


Answer



Assume that n=qm+r. Then an1=ar(am)q1, and since
am1(modam1)



we have:
(an1)(ar1)(modam1)

proving:
gcd(an1,am1)=gcd(ar1,am1).

Now, repeat. The involved exponents transform like in the computation of
gcd(n,m)=gcd(r,m)=

hence at last we get:
gcd(an1,am1)=agcd(n,m)1

qed.


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find limh0sin(ha)h without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...