Euler Totient Function

For this problem it helps to know that: 3x7x13 = 273 (a) Define the Euler Totient function, (SYMBOL) For (b) to (f) please see attached. (PLEASE SEE ATTACHMENT FOR COMPLETE PROBLEM AND PROPER SYMBOLS)

Odd Prime; Inverse; Lemma; Wilson's Theorem

Assume p is an odd prime ... *See attachment for questions

Proof about congruence modulo 43 (also expressible as equivalence modulo 43)

Let S = Z_43 (where the underscore, "_", indicates that what follows it, in this case 43, is a subscript). Let Q be a subset of S that contains ten non-zero numbers (i.e., that Q contains ten non-zero elements of S). Prove that Q contains four distinct numbers "a," "b," "c," "d" such that ab = cd in Z_43.

Context Free Grammars

Following is a big-oh relationship. Give witnesses n0 and c that can be used to prove the relationship. Choose your witnesses to be minimal, in the sense that n0 - 1 and c are not witnesses, and if d < c, then n0 and d are not witnesses. n¹º is O(3ⁿ)

Context Free Grammars / Languages from Grammars

On the ith round {see attached}, what is the length of the shortest string that is new for either of the syntactic categories? What is the length of the longest new string for: a) b)