By VICTOR SHOUP

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K − 1. Let us assume that k is chosen to be the smallest such positive integer. We claim that i = 0, or equivalently, αk = [1]n . To see this, suppose by way of contradiction that αk = αi , for some i = 1, . . , k − 1. Then we can cancel α from both sides of the equation αk = αi , obtaining αk−1 = αi−1 , and this contradicts the minimality of k. From the above discussion, we see that the ﬁrst k powers of α, that is, [1]n = α0 , α1 , . . , αk−1 , are distinct, and subsequent powers of α simply repeat this pattern.

C) If α ∈ Z∗n and γ, δ ∈ Zn , then there exists a unique β ∈ Zn such that αβ + γ = δ. 13. Verify the usual “rules of exponent arithmetic” for Zn . That is, show that for α ∈ Zn , and non-negative integers k1 , k2 , we have (αk1 )k2 = αk1 k2 and αk1 αk2 = αk1 +k2 . Moreover, show that if α ∈ Z∗n , then these identities hold for all integers k1 , k2 . 4 Euler’s phi function Euler’s phi function φ(n) is deﬁned for positive integer n as the number of elements of Z∗n . Equivalently, φ(n) is equal to the number of integers between 0 and n − 1 that are relatively prime to n.

Note that a given residue class modulo n has many diﬀerent “names”; for example, the residue class [1]n is the same as the residue class [1 + n]n . For any integer a in a residue class, we call a a representative of that class. 9. For a positive integer n, there are precisely n distinct residue classes modulo n, namely, [a]n for a = 0, . . , n − 1. Fix a positive integer n. Let us deﬁne Zn as the set of residue classes modulo n. We can “equip” Zn with binary operations deﬁning addition and multiplication in a natural way as follows: for a, b ∈ Z, we deﬁne [a]n + [b]n := [a + b]n , and we deﬁne [a]n · [b]n := [a · b]n .