MathDB

N1

Part of 2021-IMOC

Problems(1)

IMOC 2021 N1

Source: IMOC 2021 N1

8/12/2021
This problem consists of four parts.
1. Show that for any nonzero integers m,n,m,n, and prime pp, we have vp(mn)=vp(m)+vp(n).v_p(mn)=v_p(m)+v_p(n).
2. Show that if an off prime pp, a positive integer kk and integers a,ba,b satisfy p∤ ’ p∣a−bp \nmid ~^\text{'}~p|a-b and p∤kp\nmid k, then vp(ak−bk)=vp(a−b).v_p(a^k-b^k)=v_p(a-b).
3. Show that if pp is an off prime with p∣a−bp|a-b and p∤a,bp\nmid a,b, then vp(ap−bp)=vp(a−b)+1)v_p(a^p-b^p)=v_p(a-b)+1).
4. Show that if an odd prime pp, a positive integer kk and integers a,ba,b satisfy p∤a,b ’ p∣a−bp\nmid a,b ~^\text{'}~ p|a-b, then vp(ak−bk)=vp(a−b)v_p(a^k-b^k)=v_p(a-b).
Proposed by LTE.
number theory