Please do Part A,B, and C and please show step by step and explain

Hint for B: Use Proposition 5.5.20, and recall that ax 1 (mod n) means the same thing as a (mod multiplication) x = 1 for a, x Zn. You may use this fact to find an inverse for fa.

Hint for C: Use the fact that a (mod multiplication) x = 1 has no solution to show that fa is not onto, which implies that fa has no inverse.

Exercise 8.7.13. Given a number aZn, consider the function fa:Zn Zn given by fa(m)=am. (a) Show that the function f6 defined on Z7 is a bijection by finding an inverse of f6. (b) For the six numbers a=0,1,2,3,4,5 in Z6, which of these give bijections for fa ? Explain your answer. Suppose that aZn is relatively prime to n. Show that in this case, fa:ZnZn is a bijection (you may want to refer to Section 5.5.4). (*Hint*) (c) Suppose that aZn such that ax1(modn) does not have an integer solution x. Show that in this case, fa:ZnZn is not a bijection. ( Hint ) Proposition 5.5.20. Given a modular equation axc(modb), where a,b,c are integers. Then the equation has an integer solution for x if and only if c is an integer multiple of the greatest common divisor of a and b