midterm 1 solutions

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

This preview shows pages 1–2. Sign up to view the full content.

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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online