MATH 4200 Midterm: MATH 4200 LSU f07 Exam fa
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Answer each of the questions on your own paper. Be sure to show your work so that partial credit can be adequately assessed. Put your name on each page of your paper: [15 points] solve the congruence 5x 23 mod 32. Since gcd(5, 32) = 1, there is a unique solution modulo 32. Euclidean algorithm (or by inspection) 5 13 2 32 = 1, so that [5] 1. Since and are disjoint cycles, of length 5 and 4, respectively, o( ) = 5, o( ) = 4, and o( ) = lcm[5, 4] = 20. (b) find the smallest positive integer m such that m = . Since and are disjoint permutations, they commute. M = ( )m = m m = provided that m 1 mod 5 and m 0 mod 4. Solv- ing this simultaneous congruence gives m 16 mod 20.
