College Algebra Exam Review 59

College Algebra Exam Review 59 - 69 1.10. GROUPS Moreover,...

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1.10. GROUPS 69 Moreover, if p is odd and k is not divisible by p , then .1 C kp/ p s 1 . mod p s C 2 /: Hint : Use induction on s. 1.9.11. (a) Suppose that the integer a is relatively prime to the prime p . Then for all integers s ² 0 , a p s .p ± 1/ ± 1 . mod p s C 1 /: Hint : By Fermat’s little theorem, a p ± 1 D 1 C kp for some in- teger k . Thus a p s .p ± 1/ D .1 C kp/ p s . Now use the previous exercise. (b) Show that the statement in part (a) is the special case of Euler’s theorem, for n a power of a prime. 1.9.12. Let n be a natural number with prime factorization n D p k 1 1 ³³³ p k s s , and let a be an integer that is relatively prime to n . (a) Fix an index i ( 1 ´ i ´ s ) and put b D a Q j ¤ i '.p k j j / . Show b is also relatively prime to n . (b) Show that a '.n/ D b '.p k i i / , and apply the previous exercise to show that a '.n/ ± 1 . mod p k i i / . (c) Observe that a '.n/ µ 1 is divisible by p k i i for each i . Conclude that a '.n/ µ 1 is divisible by n . This is the conclusion of Euler’s theorem. 1.10. Groups An operation or product on a set G is a function from G G to
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