Explore BrainMass
Share

# Euclidean algorithm

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

1. Use the Euclidean algorithm to find
a) gcd(100, 101)
2. gcd(123, 277)
3. gcd(1529, 14039)

2. consider congruence x^2 &#8801; 16 (mod 105) for integer x
a. Give one sample solution of the congruence in the range 0&#8804;x&#8804;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&#8804;x&#8804;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&#8804;x&#8804;104 for satisfying congruence x2 &#8801; 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