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Unformatted text preview: Exam I Review Sheet Math 4023 Section 1 The flrst exam will be on Wednesday, September 22, 2010. The syllabus will be sections 1.1 and 1.2 in Lax, and the number theory handout found on the class web site, plus the handout on the method of successive squaring in computing high powers of an integer modulo n . In the following lists, pages and results on the number theory handout will be referred by preceding the number with H. Thus, Page H.23 refers to page 23 of the handout, while Proposition H.1.2.3 refers to Proposition 1.2.3 of the handout. Following are some of the concepts and results you should know: ² Know the Well-ordering principle: Any set of positive integers which has at least one element contains a smallest element. ² The cardinality of X , denoted j X j , is the number of elements of X . Some formulas for the cardinality of combinations of sets X and Y : 1. j X [ Y j = j X j + j Y j ¡ j X \ Y j . 2. j X £ Y j = j X jj Y j . 3. jP ( X ) j = 2 j X j where P ( X ) denotes the power set of X , that is, P ( X ) is the set of all subsets of X . 4. jf all functions f : X ! Y gj = j Y j j X j . ² The number of ways to choose r elements (without replacement) from an n-element set is n r ¶ = n ! r !( n ¡ r )! : ² Know the Binomial Theorem: If x and y are any numbers, then ( x + y ) n = n X k =0 x n ¡ k y k ; where n k ¶ = n ! k !( n ¡ k )! : ² Know the Division Algorithm . ² Know the deflnition of a divides b for integers a and b (notation: a j b ). ² Know the deflnition of the greatest common divisor of the integers a and b (notation: ( a; b )). ² Know the Euclidean Algorithm and how to use it to compute the greatest common divisor of integers a and b . ² Know how to use elementary row operations to codify the calculations needed for the Eu- clidean algorithm into a sequence of matrix operations as illustrated on Pages H.11 and H.12. ² Know the deflnition of relatively prime integers . ² Be sure to know Theorem H.12.3 (Page H.16) which relates the relative primeness of a and b to various divisibility conclusions. ² Know the deflnition of least common multiple of integers a and b (notation: [ a; b ]). 1 Exam I Review Sheet Math 4023 Section 1 ² Know the relationship between the greatest common divisor, least common multiple, and the product of integers a and b : ab = ( a; b )[ a; b ] : (Page H.22.) ² Know the deflnition of prime number. ² Know Euclid’s Lemma: If p is a prime, a and b are integers, and p j ab , then p j a or p j b (Corollary 2.19, Page 14). ² Know the Fundamental Theorem of Arithmetic (Theorem 2.21, Page 14 or Theorem H.1.2.7, Page H.18), and how to use it to compute the greatest common divisor and least common multiple of two integers a and b (Proposition 2.22, Page 15 or Proposition H.1.2.10, Page H.21)....
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This document was uploaded on 12/28/2011.
- Fall '09