Combinatorial and Computational Number Theory Fermat’s Little Theorem Greatest Common Divisor

See attached file for full problem description. (a) Prove that if g.c.d.(n,p) = 1,then p divides n^(p-1) –1. (b) Prove that if 3 is not a divisor of n, then 3 divides n^2 –1. (c) Prove that if 5 is not a divisor of (n – 1), 5 is not a divisor of n,and 5 is not a divisor of (n+1), then 5 divides (n^2 + 1).

Linear Congruences Set of Mutually Incongruent Solutions

Find a complete set of mutually incongruent solutions of each of the following . (a) 7x is congruent to 5 (mod 11) (b) 8x is congruent to 10 (mod 30) (c) 9x is congruent to 12 (mod 15)

Linear Congruences - The Chinese Remainder Theorem

Find all solutions of each of the systems of congruences:- (a) x is congruent to 1 (mod 2) (d) 4x is congruent to 2 (mod 6) x is congruent to 2 (mod 3) 3x is congruent to 5 (mod 7) x is congruent to 3 (mod 5) 2x is congruent to 4 (mod 11) (b) x is congruent to 1 ( mod 3) (e) x is congruent to 1 (mod 3) ... click for more

Linear Congruences - Application of Chinese Remainder Theorem

Find the least positive integer that yields the remainders 1,3 and 5 when divided by 5,7 and 9 respectively.

Fermat Numbers

The Fermat numbers are numbers of the form 2 ^2n + 1 = Φn . Prove that if n < m , then Φn │Φm – 2. The Fermat numbers are numbers of the form 2 ^2n + 1 = (Phi)n . Prove that if n < m , then (Phi)n │(Phi)m – 2.