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Math 115 - Fall 1999 - Ribet - Final

# Math 115 - Fall 1999 - Ribet - Final - 6 Find the number of...

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Math 115 Professor K. A. Ribet Final Exam December 14, 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. Each question is worth 6 points. 1 . Let n be an integer greater than 1. Let p be the smallest prime factor of n . Show that there are integers a and b so that an + b ( p - 1) = 1. 2 . Using the identity 27 2 - 8 · 91 = 1, describe the set of all integers x that satisfy the two congruences x n 35 mod 91 18 mod 27 . 3 . Let m = 2 2 3 3 5 5 7 7 11 11 . Find the number of solutions to x 2 x mod m . 4 . Calculate ± - 30 p ² , where p is the prime 101. Justify each equality that you use. 5 . Write 2 + 8 as an inﬁnite simple continued fraction.
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Unformatted text preview: 6 . Find the number of primitive roots mod p 2 when p is the prime 257. 7 . Express the continued fraction h 6 , 6 , 6 , . . . i in the form a + b √ d , with a and b rational numbers and d a positive non-square integer. 8 . Suppose that p = a 2 + b 2 , where p is an odd prime number and a is odd. Show that ± a p ² = +1. (Use the Jacobi symbol.) 9 . Let n be an integer. Show that n is a diﬀerence of two squares (i.e., n = x 2-y 2 for some x, y ∈ Z ) if and only if n is either odd or divisible by 4. 10 . Let n be an integer greater than 1. Prove that 2 n is not congruent to 1 mod n ....
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