1. a) (a) Use the Fermat factoring method to factorise 4423787. [You may use a calculator but all the steps must be written down in the solution.] b) Compute 58839154 (mod 4423787). 2. a) You are given that p = 4 is a prime number and that 3p-4 = 1 (mod p). Find all values p with such properties. b) Let a, b, n be integer numbers such that gcd(a, b) = 1. Show that: if a | n and b | n then ab | n. You may use any results which were proven in lectures. c) Find an integer number n such that 5n is a fifth power of some integer number, 6n is a sixth power of some integer number and 7n is a seventh power of some integer number. [Hint: Consider the prime factorisation of n. If your answer is huge you do not have to write its decimal expansion. You may leave it as a mathematical expression.] Document Preview:

a) (a) Use the Fermat factoring method to factorise 4423787. [You may use a calculator but all the steps must be written down in the solution.] b) Compute 58839154 (mod 4423787). a) You are given that p = 4 is a prime number and that 3p-4 = 1 (mod p). Find all values p with such properties. b) Let a, b, n be integer numbers such that gcd(a, b) = 1. Show that: if a | n and b | n then ab | n. You may use any results which were proven in lectures. c) Find an integer number n such that 5n is a fifth power of some integer number, 6n is a sixth power of some integer number and 7n is a seventh power of some integer number. [Hint: Consider the prime factorisation of n. If your answer is huge you do not have to write its decimal expansion. You may leave it as a mathematical expression.]

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