[理工] [離散] 整係數方程式

[movo11 (Larry)]

True of False:
1.Let n be a fixed positive integer that satisfies the property:
  For all a,b∈Z+, if n|ab then n|a or n|b. Then n=1 or n is prime.
2.With gcd(a,b)=1, if a|c and b|c are true, then ab|c is true.


Show the following:
1.gcd(m,n) is a divisor of n-m.


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◆ From: 123.192.157.110
推 ddczx:1.T 2.F  令gcd(m,n)=d, d|m,d|n 故d|n-m
→ movo11:第一題true怎麼推導? 還有第二題是true喔
推 ddczx:看漏gcd(a,b)=1了...  1. 證 p->q = 證 ~q -> ~p
→ ddczx:設n非1or質數,則n=xy for some x,y=/=1,此時令a=x,b=y得矛盾
推 ddczx:2.gcd(a,b)=1,故a,b最小公倍數是ab,又a|c,b|c ,c是a,b之
→ ddczx:公倍數為最小公倍數ab之倍數,故ab|c
推 a00000816:令gcd(m,n)=k,k|m&k|n,令m=p*k,n=q*k(p,q∈Z),欲證k|n-m
→ a00000816:->k|(q*k-p*k)->k(q-p),故k|n-m
推 a00000816:阿沒看到樓上已經解了XD
推 a1098137129:可是第一題假如說代數字進去 a=6 b=12 n=6
→ a1098137129:這樣答案是F吧? 還是我題目有看漏的?