sol7 - Math 465 Solution Set 7 1. Find the order of 2, 3,...

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Math 465 Solution Set 7 Ae Ja Yee 1. Find the order of 2 , 3 , and 5 modulo 23. Solution. Since 23 is prime, φ (23) = 22, which has proper divisors 1 , 2 , 11. 2 1 6≡ 1 , 2 2 = 4 6≡ 1 , 2 11 = 2 1+2+8 = 2 · 2 2 · 2 8 2 · 4 · 3 24 1 (mod 23) , 3 1 6≡ 1 , 3 2 = 9 6≡ 1 , 3 11 = 3 1+2+8 = 3 · 3 2 · 3 8 3 · 9 · 6 162 1 (mod 23) , 5 1 6≡ 1 , 5 2 = 25 6≡ 1 , 5 11 = 5 1+2+8 = 5 · 5 2 · 5 8 5 · 2 · 16 160 ≡ - 1 6≡ 1 (mod 23) . Thus 2 and 3 have order 11 and 5 has order 22. 2 2. Prove that φ (2 n - 1) is a multiple of n for any n > 1. (Hint: The integer 2 has order n modulo 2 n - 1.) Solution. It is clear that for n > 1, 2 n 1 (mod 2 n - 1) , 2 k 6≡ 1 (mod 2 n - 1) for any k < n. So the order of 2 modulo 2 n - 1 is n . By Euler’s theorem, we know 2 φ (2 n - 1) 1 (mod 2 n - 1) . Therefore, the order of 2 divides φ (2 n - 1), that is n | φ (2 n - 1). 2 3. Let a has order 3 modulo p . (a) Show that
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sol7 - Math 465 Solution Set 7 1. Find the order of 2, 3,...

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