Given a number a Zn, consider the function fa : Zn Zn given by...

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Given a number a € Zn, consider the function fa : Zn → Zn given by fa(m) = ao m. (a) Show that the function fe defined on Z7 is a bijection by finding an inverse of fe. (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 a € Zn is relatively prime to n. Show that in this case, fa: Zn → Zn is a bijection (c) Suppose that a € Zn such that ar = 1 (mod n) does not have an integer solution z. Show that in this case, fa: Zn→ Zn is not a bijection. Given a modular equation ar = c (mod b), where a,b,c are integers. Then the equation has an integer solution for z if and only if c is an integer multiple of the greatest common divisor of a and b. Given a number a € Zn, consider the function fa : Zn → Zn given by fa(m) = ao m. (a) Show that the function fe defined on Z7 is a bijection by finding an inverse of fe. (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 a € Zn is relatively prime to n. Show that in this case, fa: Zn → Zn is a bijection (c) Suppose that a € Zn such that ar = 1 (mod n) does not have an integer solution z. Show that in this case, fa: Zn→ Zn is not a bijection. Given a modular equation ar = c (mod b), where a,b,c are integers. Then the equation has an integer solution for z if and only if c is an integer multiple of the greatest common divisor of a and b.

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Lets tackle each part of the question one by one a Show that the function f6 defined on Z7 is a bijection by finding an inverse of f6 To prove that f6 ... View the full answer