Summary: Congruence
Putnam practice
November 12, 2003
We say a is congruent to b modulo n and write a b(mod n) if n|(a-b).
Let p be a prime and Zp denote the set {0, 1, ...p - 1}. Define + and · on Zp
using congruence modulo p. The system (Zp, +, ·) is a finite field.
Example 1 Prove that 3636 + 4141 is divisible by 77.
Solution: Note 41 -36(mod 77). Thus
3636
+ 4141
3636
+ (-36)41
3636
(1 - 365
)(mod 77)
Also note
36 1(mod 7)
and
365
35