1. Prove that there are infinitely many prime numbers. 2. Show that the gcd operator is…

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1. Prove that there are infinitely many prime numbers.

2. Show that the gcd operator is associative. That is, for all integers m, n, and h, we have gcd (m, gcd(n, h)) = gcd (gcd(m, n),h).

3. Prove that if m and n are both odd, then gcd (m, n) = gcd ((m − n)/2, n).

4. Find the necessary condition to have equation mx ≡ my mod n imply x ≡ y mod n. 59. Assuming that p is a prime number, find the solutions of the equation x 2 = [1]p .

4. In an RSA cryptosystem, let p and q be the large primes, let n = pq, and let pub be the public key. Show that pub (a)pub(b) is congruent to pub(ab) modulo n

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