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Show that (1 + p)p ≡ 1. p2 More generally, show by Induction that for n pn−2 (1 + p) ≡ 1 + pn−1 , pn 2, the following congruences are true: n−1 (1 + p)p ≡ 1. pn What can you say about the case p = 2? 1-15. 14160, 51/11, 1725/1193, 1193/1725, −1193/1725, 30031/16579, 1103/87. In each case determine all the convergents. 1-16. If n is a positive integer, what are the continued fraction expansions of −n and 1/n? What about when n is negative? ] Try to find a relationship between the continued fraction expansions of a/b and −a/b, b/a when a, b are non-zero natural numbers.

D|n Then ψ is multiplicative. Proof. θ = ψ ∗ η, so by M¨obius Inversion, ψ = θ ∗ µ, implying that ψ is multiplicative. 52 3. ARITHMETIC FUNCTIONS Problem Set 3 3-1. Let τ : Z+ −→ R be the function for which τ (n) is the number of positive divisors of n. a) Show that τ is an arithmetic function. b) Suppose that n = pr11 pr22 · · · prt t is the prime power factorization of n, where 2 p1 < p2 < · · · < pt and rj > 0. Show that τ (pr11 pr22 · · · prt t ) = (r1 + 1)(r2 + 1) · · · (rt + 1). c) Is τ multiplicative?

2 1 3 We can calculate the composition τ ◦σ of two permutations τ, σ ∈ Sn , where τ σ(k) = τ (σ(k)). Notice that we apply σ to k first then apply τ to the result σ(k). For example, 1 2 3 3 2 1 1 2 3 3 1 2 = 1 2 3 , 1 3 2 1 2 3 2 3 1 1 2 3 2 3 1 1 2 3 3 1 2 1 2 3 3 1 2 = 1 2 3 1 2 3 = ι. In particular, = −1 . Let X be a set with exactly n elements which we list in some order, x1 , x2 , . . , xn . Then there is an action of Sn on X given by σ · xk = xσ(k) (σ ∈ Sn , k = 1, 2, . . , n). For example, if X = {A, B, C} we can take x1 = A, x2 = B, x3 = C and so 1 2 3 · A = B, 2 3 1 1 2 3 · B = C, 2 3 1 1 2 3 · C = A.