Math 115 - Fall 2000 - Ribet - Midterm 2

Math 115 - Fall 2000 - Ribet - Midterm 2 - . If p is a...

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Math 115 Professor K. A. Ribet Last Midterm Exam November 3, 2000 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 (3 points) . Calculate the number of primitive roots mod 35035 = 5 · 7 2 · 11 · 13. 2 (6 points) . What is the remainder when one divides the prime number 1234567891 by 11? What is the remainder when 11 1234567890 is divided by 1234567891? 3 (5 points) . Find a mod 29 inverse to the 2 × 2 matrix ± 1 2 3 9 ² mod 29. 4 (9 points)
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Unformatted text preview: . If p is a prime, show that all prime divisors of 2 p-1 are congruent to 1 mod p . (For example, 2 11-1 = 23 89 is divisible by the primes 23 and 89 and by no others.) 5 (7 points) . Let be the Mobius function, and let be the function whose value on n 1 is the number of divisors of n . Explain why the function F ( n ) := X d | n ( d ) ( d ) satises the relation F ( n 1 n 2 ) = F ( n 1 ) F ( n 2 ) when gcd( n 1 , n 2 ) = 1. Calculate F ( p e ) when p is a prime and e is a positive integer....
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