Midterm 2 - 2 n + 1, with n 1. Note that p 1 mod 4. With...

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MATH 115 MID-TERM 2 1. a) ( 2 pts ) Find the number of solutions to the congruence, if any: x 13 2 mod 53 b) ( 2 pts ) Find the number of solutions to the congruence, if any: x 20 13 mod 61 c) ( 2 pts ) Let p be a prime, with p 2 mod 3. Show that, for an integer a with ( a,p ) = 1, the congruence x 3 a mod p always has a unique solution. 2. a) ( 2 pts ) Show by induction that 2 2 n 2 or 4 mod 7 for integer n 1. b) ( 2 pts ) Use part a) to show that ± 2 2 n + 1 7 ² = - 1 c) ( 2 pts ) Suppose that p is a prime of the form 2
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Unformatted text preview: 2 n + 1, with n 1. Note that p 1 mod 4. With this assumption, show that 7 ( p-1) / 2 -1 mod p 3. a) ( 2 pts ) Let p be an odd prime, with p 6 = 5. Show that -5 p = -1 p p 5 b) ( 4 pts ) Hence nd the congruence condition on an odd prime p 6 = 5, for which x 2 -5 mod p is solvable. 4. ( 2 pts ) Determine whether x 2 35 mod 1003 is solvable (given that 1003 is a prime). 1...
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