# Euclidean algorithm

1. Use the Euclidean algorithm to find

a) gcd(100, 101)

2. gcd(123, 277)

3. gcd(1529, 14039)

2. consider congruence x^2 ≡ 16 (mod 105) for integer x

a. Give one sample solution of the congruence in the range 0≤x≤104 that is different from 4 and also from 101

b. Find the number of solutions of ths congruence among integers in the range 0≤x≤104 by resorting to the Chiness Reminder Theorem

Clarification: you are not asked to list all solutions. In particular, a direct inspection of each number x such that 0≤x≤104 for satisfying congruence x2 ≡ 16 (mod 105).

3. Use mathematical induction to show that 3 divides n^3 + 2n, whenever n is a natural number.

4. Let fn denote the n-th Fibonacci number. Use mathematical inducation to show that

fn+1fn-1 - f2n = (-1)2

for all positive integers n

https://brainmass.com/math/discrete-structures/euclidean-algorithm-270378

#### Solution Summary

Euclidean algorithm is applied.