# Homework Help: A couple of Number Theory questions

1. May 9, 2012

### squire636

1. Find all solutions x (with 0 ≤ x ≤ 96) to the congruence 13x^385 + 73x^304 + x^290 + 10x^193 + 24x^112 + 70x + 76 ≡ 0 (mod 97)

I was able to reduce, using Fermat's Little Theorem, to get 97x^16 + x^2 + 93x + 76 ≡ 0 (mod 97), but I don't know how to proceed from there. Is there another trick I can use?

2. http://imgur.com/a/DUyHC

RR denotes two adjacent quadratic residues, while NN denotes two adjacent quadratic non-residues. RN is a residue followed by a non-residue, and NR is a non-residue followed by a residue. I tried to solve the problems by adding and subtracting the four expressions given in part b but I haven't made any progress.

Thanks for the help!

Last edited: May 9, 2012
2. May 9, 2012

### SammyS

Staff Emeritus
One quick question: What is 97 mod 97 ?

3. May 9, 2012

### squire636

One quick response: UGHHHH sometimes I'm an idiot. Thanks!

4. May 9, 2012

### SammyS

Staff Emeritus
Did you find the solution ?

5. May 9, 2012

### squire636

I sure did, I found two solutions just by using the quadratic formula on the equation that remains after the x^16 term goes to zero. Then I just had to make sure that they're between 0 and 97. My solutions aren't integers, which is sort of frustrating, but oh well.

Any advice on the other problem?

6. May 9, 2012

### SammyS

Staff Emeritus
Well, there are two integer solutions.

What numbers are congruent to 93 and/or 76 (mod 97) ?

Last edited: May 9, 2012
7. May 9, 2012

### squire636

Oh good call, thanks so much. I got integer solutions once I changed the x and the constant term to congruent values.