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Math 115 - Spring 1998 - Ribet - Midterm 1

# Math 115 - Spring 1998 - Ribet - Midterm 1 - k(0 ≤ k ≤...

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Math 115 Professor K. A. Ribet First Midterm Exam February 25, 1998 Instructions: Answer question #2 and three other questions. 1 (6 points) . Find all solutions to the congruence x 2 p mod p 2 when p is a prime number. 2 (9 points) . Using the equation 7 · 529 - 3 · 1234 = 1, find an integer x which satisfies the two congruences x 123 mod 529 321 mod 1234 and an integer y such that 7 y 1 mod 1234. (No need to simplify.) 3 (7 points) . Suppose that p is a prime number. Which of the p + 2 numbers ( p + 1
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Unformatted text preview: k ) (0 ≤ k ≤ p + 1) are divisible by p ? [Example: The seven binomial coeﬃcients ( 6 k ) are 1, 6, 15, 20, 15, 6, 1; the middle three are divisible by 5.] 4 (7 points) . Let p be a prime and let n be a non-negative integer. Suppose that a is an integer prime to p . Show that b := a p n satisﬁes b ≡ a mod p and b p-1 ≡ 1 mod p n +1 . 5 (6 points) . Show that n 4 + n 2 + 1 is composite for all n ≥ 2....
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