# Modulo and congruencies in class

1. Jun 27, 2004

### 1+1=1

i am learning about modulo and congruencies in class and i am seeking some help.

i need to find a complete residue system mod 11 consisting of odds only.

show that every pos int. n, 7^n congruent to 1+6n (mod36)

find the least residue of (n-)! mod n for several values of n. find a rule but no need for a proof.

here is what i know so far...

with the least residue problem, i know that a=mq+r w/ 0<=r<1 then r is the least residue, so it is like the remainder correct? any one offer further advise to help w/ this?

to find out the conplete resideu system of mod 11 means that m divides (a-b) where a and b are congruent to each other. any other help?

the second problem i really don't know how to do but would like help! please.

2. Jun 27, 2004

### Muzza

Try induction. Or maybe you could write 7^n as (1 + 6)^n and expand it with the binomial theorem...

So you need to 10 find odd integers that are congruent to 0, 1, ..., 10 modulo 11. The first one is simple, since 11 = 0 (mod 11). The numbers which are congruent to 1 mod 11, are 1 more than a multiple of 11, i.e they are of the form 1 + 11k. Should be easy from there.

3. Jun 27, 2004

### 1+1=1

oooo thanks muzza! i should use the weak form of induction, that should take care of it right? choose 1 and show it holds then sub in k for n and then k+1, then go from there? wooo!!

then the seconnd one i just need to use that formula to get my odd numbers mod 11? muzza you are a LIFESAVER ! (no you are really)!