Math 115 - Fall 1999 - Ribet - Midterm 1

Math 115 - Fall 1999 - Ribet - Midterm 1 - (i) gcd( a, b )...

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Math 115 Professor K. A. Ribet First Midterm Exam September 23, 1999 This is a closed-book exam: no notes, books or calculators are allowed. Explain your answers in complete English sentences. No credit will be given for a “correct answer” that is not explained fully. 1 (4 points) . Find the remainder when 2 33 is divided by 31. 2 (4 points) . Use the identity 27 2 - 8 · 91 = 1 to find an integer x such that 27 x = 14 mod 91. 3 (4 points) . Find all prime numbers p such that p 2 + 2 is prime. 4 (5 points) . Suppose that ax + by = 17, where a , b , x and y are integers. Show that the numbers gcd( a, b ) and gcd( x, y ) are divisors of 17. Decide which, if any, of the following four possibilities can occur:
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Unformatted text preview: (i) gcd( a, b ) = gcd( x, y ) = 1; (ii) gcd( a, b ) = 17 and gcd( x, y ) = 1; (iii) gcd( a, b ) = 1 and gcd( x, y ) = 17; (iv) gcd( a, b ) = gcd( x, y ) = 17. 5 (6 points) . Suppose that n is composite: an integer greater than 1 that is not prime. Show that ( n-1)! and n are not relatively prime. Prove that the congruence ( n-1)! -1 mod n is false. 6 (6 points) . Prove that-1 is not a square modulo the prime p if p 3 mod 4. 7 (6 points) . Show that x 8 1 mod 20 if x is an integer that is prime to 20. Find the integer t such that t 9 = 760231058654565217 7 . 60231 10 17 ....
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