4023f11ex1reva

# 4023f11ex1reva - Exam I Review Sheet Solutions Math 4023...

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Exam I Review Sheet Solutions Math 4023 The ﬂrst exam will be on Friday, September 23, 2011. The syllabus will be sections 0.1 through 0.4 and 0.6 in Nagpaul and Jain, and the corresponding parts of the number theory handout found on the class web site. In the following lists, pages and results on the number theory handout will be referred by preceding the number with H. Following are some of the concepts and results you should know: 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 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: gcd( 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 done in class and illustrated on Pages H.11 and H.12. ² Know the deﬂnition of relatively prime integers . ² Know the deﬂnition of least common multiple of integers a and b (notation: [ a; b ]). ² Know the deﬂnition of prime number. ² Know what it means for an integer a to be congruent modulo n to another integer b (notation a · b mod n ). ² Know the deﬂnition of congruence class of a modulo n (notation [ a ] n ). ² Know the deﬂnition of the number system Z n , and how to do arithmetic in Z n : [ a ] n + [ b ] n = [ a + b ] n [ a ] n [ b ] n = [ ab ] n ² Know the deﬂnition of [ a ] n is invertible in Z n , and know the criterion of invertibility of [ a ] n : An element [ a ] n 2 Z n is invertible (or has a multiplicative inverse) if and only if gcd( a; n ) = 1, that is, if and only if a and n are relatively prime. Moreover, if r and s are integers such that ar + ns = 1, then [ a ] ¡ 1 n = [ r ] n . (Theorem 0.2.8 (text) and Proposition H.1.4.5, Page H.38.) 1

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Exam I Review Sheet Solutions Math 4023 ² Know how to use the Euclidean algorithm to compute [ a ] ¡ 1 n , when the inverse exists. ² Z / n is the set of invertible elements of Z n . ² Z / n is closed under multiplication, i.e., the product of any two elements of Z / n is in Z / n . ² If p is a prime, then Z / p = Z p n f [0] n g . (Corollary H.1.4.6, Page 39). ² Know the Chinese Remainder Theorem (Theorem 0.6.4), and how to solve simultaneous congruences. ² The Euler phi-function at n , denoted ( n ), is deﬂned to be the number of positive integers less than or equal to n which are relatively prime to n . It is also true that ( n ) = j Z / n j .
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4023f11ex1reva - Exam I Review Sheet Solutions Math 4023...

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