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    Euclidean algorithm

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    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

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    Solution Summary

    Euclidean algorithm is applied.