midterm 1 solutions

midterm 1 solutions - Math 55 First Midterm Exam...

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Math 55, First Midterm Exam SOLUTIONS (1) There are twelve positive integers less than 36 that are relatively prime to 36. Hence φ (36) = 12 . Since 37 is a prime number, we have φ (37) = 37 1 = 36 . In the second question of Quiz # 2 we saw that φ ( p k ) = p k p k - 1 for any power of a prime number p . Therefore, φ (81) = φ (3 4 ) = 3 4 3 3 = 54 and φ (1024) = φ (2 10 ) = 2 10 2 9 = 512 . (2) It is important to note that the domain is the set of nonnegative integers. (a) The statement x (( x 2 < 10) ( | 3 x | > 2)) is true because x = 0 satisFes both inequalities in the conjunction. (b) The statement x (( x n = 4) ( x 5 > 1)) is false . It does not hold for x = 3. (c) The statement x y ( x + y = 0) is false because positive integers have no additive inverses among the nonnegative integers. (d) The statement x y ( xy = 0) is true because x = 0 is a nonnegative integer, and it satisFes 0 · y = 0 for all y .
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midterm 1 solutions - Math 55 First Midterm Exam...

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