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312f10mid1

# 312f10mid1 - -1=0 1-4 6-4 1 = 0 Prove that the alternating...

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My NAME is: Problem 1 2 3 4 5 Total Score MAT 312 Applied Algebra Midterm 1 October 5, 2010 No books or notes may be consulted during this test. Explain your answers carefully. Total score = 100. 1. (15 points) Prove that 17 is not rational, by showing that it cannot be written as p/q with p and q integers, ( p,q ) = 1. You may use the fact that every integer can be uniquely (up to order) written as a product of primes. 2. (20 points) Observe that 1 = 1 2 , 1 + 3 = 2 2 , 1 + 3 + 5 = 3 2 . Use induction to prove that the sum of the first n odd numbers is equal to n 2 , i.e. 1 + 3 + · · · + (2 n - 1) = n 2 . 3. (15 points) Observe that 1 - 1 = 0, 1 - 2+1 = 0, 1 - 3+3 - 1=0, 1 - 4+6 - 4+1 = 0. Prove that the alternating sum of the elements in the
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Unformatted text preview: -1=0, 1-4+6-4+1 = 0. Prove that the alternating sum of the elements in the n-th row of Pascal’s triangle is equal to 0, i.e. p n P-p n 1 P + p n 2 P- ··· ± p n n P = 0 . 4. (a) (10 points) ±ind the greatest common denominator d of 731 and 645. (b) (10 points) Express d in the form 731 j + 645 k , with j and k integers. (c) (10 points) Give prime factorizations for 81 , 82 , 83. 5. (a) (10 points) Calculate the multiplicative inverse of 19 modulo 21. (b) (10 points) Explain why 14 does not have a multiplicative inverse modulo 21. END O± EXAMINATION 1...
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