n1 - Introduction to Number Theory Notes 1 Divisibility in...

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Unformatted text preview: Introduction to Number Theory Notes 1. Divisibility in Z Deﬁnition 1.1 Let a and b are integers with a = 0. We say that “ a divides b” if there is an integer c such that b = ac. Terminology: “a is a divisor of b”, “a is a factor of b,” “b is a multiple of a” Notation: a|b Theorem 1.1 i. a|a for every non-zero integer a. ii. If a and b are non-zero integers such that a|b and b|a, then a = ±b. iii. If a, b, c are non-zero integers such that a|b, b|c, then a|c. Proof. i. a = 1 · a. ii. Since a|b and b|a, there are e, f ∈ Z such that ae = b and bf = a. Therefore, a = bf = aef and, since a = 0, ef = 1. Therefore, e = f = ±1 and a = ±b. iii.Since a|b and b|c, there are e, f ∈ Z such that ae = b and bf = c. Therefore c = bf = aef = a(ef ) and a|c. Theorem 1.2 If a, b, k, l ∈ Z and c|a, c|b, then c|(ka + lb). Proof. Since c|a and c|b, there are e, f ∈ Z such that ce = a and cf = b. Therefore, ka + lb = c(ke + lf ) and c|(ka + lb). Theorem 1.3 (Division Algorithm) If a, b ∈ Z and b > 0, then there are unique integers q and r such that a = bq + r with 0 ≤ r < b. Proof. Existence: Let S be the set of non-negative integers of the form a − bk for some k ∈ Z. • S = ∅ : For instance, for any integer q ≤ a − 1, we have b a a − bq ≥ a − b( b − 1) ≥ 0. Therefore it has a least element r = a − bq, (q ∈ Z). • By construction, r ≥ 0. • r < b : Suppose that r ≥ b. Then a − b(q + 1) = r − b ≥ 0, hence a − b(q + 1) ∈ S. However, a − b(q + 1) = r − b < r. This contradicts the choice of r as least element of S. Therefore, the r and corresponding q just constructed satisfy the requirements set in the theorem. Uniqueness: Suppose there are pairs (q1 , r1 ) and (q2 , r2 ) such that: a = bq1 + r1 and 0 ≤ r1 < b a = bq2 + r2 and 0 ≤ r2 < b Upon subtraction of the two equations, we have 0 = b(q1 − q2 ) + (r1 − r2 ) or r2 − r1 = b(q1 − q2 ). This implies b|(r2 − r1 ). Since 0 ≤ r1 < b and 0 ≤ r2 < b, −b < r2 − r1 < b. Therefore, r2 − r1 must be 0. This, in turn, implies that a = bq1 + r1 and a = bq2 + r1 and hence, q1 = q2 . ...
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n1 - Introduction to Number Theory Notes 1 Divisibility in...

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