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College Algebra Exam Review 57

# College Algebra Exam Review 57 - Example 1.9.21 Take n D 7...

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1.9. COUNTING 67 Proof. The formula in part (a) was obtained previously. The result in part (b) is obtained by rewriting n.1 ± 1 p 1 /.1 ± 1 p 2 / ²²² .1 ± 1 p s / D p k 1 1 .1 ± 1 p 1 /p k 2 2 .1 ± 1 p 2 / ²²² p k s s .1 ± 1 p s / D '.p k 1 1 / ²²² '.p k s s /: n Corollary 1.9.19. If m and n are relatively prime, then '.mn/ D '.m/'.n/ . The following theorem of Euler is a substantial generalization of Fer- mat’s little theorem: Theorem 1.9.20. (Euler’s theorem). Fix a natural number n . If a 2 Z is relatively prime to n , then a '.n/ ³ 1 . mod n/: One proof of this theorem is outlined in the Exercises. Another proof is given in the next section (based on group theory).
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Unformatted text preview: Example 1.9.21. Take n D 7 and a D 4 . '.7/ D 6 , and 4 is relatively prime to 7, so 4 6 ³ 1 . mod 7/ . In fact, 4 6 ± 1 D 4095 D 7 ´ 585 . Take n D 16 and a D 7 . '.16/ D 8 , and 7 is relatively prime to 16, so 7 8 ³ 1 . mod 16/ . In fact, 7 8 ± 1 D 5764801 D 16 ´ 360300 . Exercises 1.9 1.9.1. Prove the binomial theorem by induction on n . 1.9.2. Prove that 3 n D P n k D ± n k ² 2 k . 1.9.3. Prove that 3 n D P n k D . ± 1/ k ± n k ² 4 n ± k ....
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