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4 marks iii let φ x be the polynomial x 4 x 3 x 2 x

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(4 marks) (iii) Let Φ( X ) be the polynomial X 4 + X 3 + X 2 + X + 1 . (a) Show that the set of primes p for which there is an integer n with p | Φ( n ) is infinite. (3 marks) (b) Now let a be an integer, and let p > 5 be a prime dividing Φ( a ) . Show that p divides a 5 - 1 , and deduce that the residue class of a in ( Z /p Z ) × , the unit group of Z /p Z , has order 5 . (5 marks) (c) Now deduce that 5 | p - 1 , and conclude that there are infinitely many primes of the form 5 n + 1 . (3 marks) 3 (i) Show that b x + y c - b x c - b y c = 0 or 1 for all x, y R . (Recall that b x c is the greatest integer not exceeding x. ) (3 marks) Now let n be a positive integer, and let p be a prime. (a) Write down a formula for the highest power of p that divides n ! , and calculate the number of zeros at the end of 2009! . (4 marks) (b) Show that if p r divides 2 n n then p r 2 n. (4 marks) (c) Show that if n 3 and 2 n/ 3 < p n, then 2 n n is not divisible by p. (4 marks) (ii) State Bertrand’s Postulate. (1 mark) Prove the following statements. (a) If n is a positive integer then the binomial coefficient 2 n n is not a square. (3 marks) (b) Call a positive integer n a primeone if either n = 1 or n is a prime number. Then every positive integer can be written as a sum of distinct primeones.
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