## Description

Math 422: Introduction to Number Theory

Homework on 9

A. Write a program that takes as input positive integers n and b, and returns

n in base b. The output can be a list of digits. You may assume b ≤ 10.

B. Silverman 9.1.

C. Silverman 9.2.

D. Silverman 10.2.

E. Let p be a prime, and suppose gcd(a, p) = 1. Show that if ax ≡ c (mod p),

then x ≡ cap−2

(mod p).

F. Suppose gcd(x, 97) = 1 and x

n ≡ 1 (mod 97), where 1 ≤ n ≤ 96. Show

that n | 96.

G. Let p(x) = x

33 − x. Show that if n is an integer, then 15 | p(n).

H. Suppose a, n are integers with n 6= 0 and gcd(a, n) 6= 1. Show that a

r 6≡ 1

(mod n) for any positive r.

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