UMBC CMSC203 Discrete Structures, Section 06, Spring 2016

Homework 6, Due Thursday, 03/10

1. Arithmetic with mod. Use the repeated squaring technique to compute 26³⁷ % 77. Show all of your work. Your work should not have any numbers bigger than 77² = 5929.

2. Inverses mod 23. For each integer x, 1 ≤ x < 23, find an integer y, 1 ≤ y < 23, such that

   x ⋅ y ≡ 1 (mod 23).

   In other words, x ⋅ y % 23 = 1. Then, x and y are called inverses modulo 23.

3. GCD Proof. Let a and b be integers such that a is even and b is odd. Argue that gcd(a, b) = gcd(a/2, b).

4. Euclid's Algorithm. Use Euclid's algorithm to compute gcd(18893511, 1154300). Show all of your work.

5. Extended Euclid's Algorithm. Use the Extended Euclid's Algorithm to find the multiplicative inverse of 173 modulo 235012. Show all of your work.

   The Extended Euclid's Algorithm is in Chapter 8 of Epp 4/e and Chapter 10 of Epp 3/e. See also Notes on the Extended Euclid's Algorithm.