Math 115 - Spring 1994 - Ribet - Midterm 1

Math 115 - Spring 1994 - Ribet - Midterm 1 - →∞ n k 1...

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Math 115 Professor K. A. Ribet First Midterm Exam, February 22, 1994 1. Is 270 a square modulo the prime number 691? 2. The decimal expansion of 1 / 7 is 0 . 142857 = . 142857142857 .. . . Find all prime numbers p for which the decimal expansion of 1 /p has period six. [It may help to know that 99 = 3 2 · 11, 999 = 3 3 · 37, 9999 = 3 2 · 11 · 101, 99999 = 3 2 · 41 · 271, and 999999 = 3 3 · 7 · 11 · 13 · 37.] 3. Using the Euclidean algorithm, find integers n and m such that 13 n + 47 m = 1. 4. Let n be a positive integer. Calculate the limit lim k
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Unformatted text preview: →∞ n k +1 σ ( n k ) , where σ denotes, as usual, the function whose value at m is the sum of the divisors of m . 5. Show that there are an infinite number of primes which are congruent to 7 mod 8. [If P 1 ,. .. ,P n are such primes, consider ( P 1 ··· P n ) 2-2.] 6. Let n be a positive integer. Show that 2 n ≡ 1 mod n if and only if n = 1. [For n > 1, consider the situation modulo the smallest prime number dividing n .]...
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This note was uploaded on 10/31/2009 for the course STAT 131A taught by Professor Isber during the Spring '08 term at Berkeley.

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