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1. t c ≡m a ◦ b 2. By Property 1, it is clear that ≡m is an equivalence relation over Zm which is preserved under modular addition, subtraction and multiplication. The next thing that comes to the mind is division. The modular counterpart of division is called a ’multiplicative inverse’. 3. 3 Given integers a, m, an integer b is the multiplicative inverse of a modulo m if ab ≡m 1. We say that a−1 = b. Note that a multiplicative inverse need not exist for any arbitrary integer a. For example, 2 doesn’t have a multiplicative inverse modulo 4.

5 tailers expansion of f (x + h) is f (x) + hf (x) + h2 2! f (x) + · · · + hn n n! f (x), as f t (x) = 0 when t > n. 6 solving f (x) ≡pα 0 Proof: if r is a solution to f (x) ≡pα 0 then f (r) ≡pt 0 for t = 1, 2, . . , α. ji consider α ≥ 2. if there is a solution uiα of f (x) ≡pα 0 then there is solution uα−1 of f (x) ≡pα−1 0 such that ji α−1 i for some integer v. 3) i but f (ujα−1 ) ≡pα−1 0. 4 we can find all the solutions of v and then ujα−1 +vpα−1 α will be solutions of f (x) ≡p 0 i . it only means that there are no some times it may happen that there are no v corresponding to some ujα−1 ji solutions of f (x) ≡pα 0 arising from this particular uα−1 .

19) (p − 1)! ap−1 p−1 a ap ≡p ≡p ≡p Note that when we vary i in the LHS of Eq. 18, we get a different value of j each time. This accounts for the (p − 1)! term in the RHS of subsequent equations. 9 If ap ≡q a and aq ≡p a where p = q are primes, then apq ≡pq a. 4. 27) ✷ 56CHAPTER 10. 1 Introduction In this lecture, we will discuss Euler s Theorem, Generalisation of Fermat Little Theorem and Chinese Remainder Theorem. 2 EULER s PHI-FUNCTION For n ≥ 1, The number φ(n) denote the number of postive integer not exceeding n , that are relatively prime to n.

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