The subject is discrete math, one of the subjects on the advanced math. This is all I was given. I`m working on these problems too, and expecting the expert-answers by the time I finish, so that I can correct my answers. Thanks a lot!

1) The Fibonacci Sequence is dened as f1 = 1, f2 = 1 and fn+i = fn+f1Ila so that the sequence proceeds 1,1,23,53,13,... Prove, by induction, that 2122: fnfn+1 1.=1 2) a) Let m and n be relatively prime integers. Use Bezout's identity to prove that m has a multiplicative inverse modulo n 1:) Use the fundamental theorem of arithmetic to prove that if m and n are relatively prime integers and n divides ml: for some integer k, then 12 divides k (This is a generalization of Euclid's lemma) c) Use the result from part b) to prove that if m and n are relatively prime integers then the multiplicative inverse of m modulo 7: is unique. 3) The Chinese remainder theorem guarantees a solution to the system of congruences m E 3 (mod 5) m E 2 (mod 8) m E 5 (mod 7) Find the smallest positive solution. 4) Let p be a prime. Prove that 1 X 2 x 3... X (p 1) E 1(modp)