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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 Fall '08
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 Algebra, Mathematical Induction, Counting, Trigraph, 1 k, Prime number, 3 k

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