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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