ex2solsSummer09 - Math 347 C1 HOUR EXAM II 21 July 2009...

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

View Full Document Right Arrow Icon

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

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

Unformatted text preview: Math 347 C1 HOUR EXAM II 21 July 2009 SOLUTIONS 1. Consider the equation: 5 x + 2 y = 37 . a) Find integers x , y , which satisfy the equation. b) Find integers x , y , with y positive, which satisfy the equation. (If your answer to part a) has y > 0, then you are done.) SOLUTION a) Since 5 and 2 are relatively prime, there are integers m and n such that 5 m + 2 n = 1. By inspection, m = 1 and n =- 2 work, i. e. (5 × 1) + (2 × (- 2)) = 1 . Then we can multliply this equation by 37 to get (5 × 37) + (2 × (- 2)37) = 37 , that is (5 × 37) + (2 × (- 74)) = 37 . Thus, x = 37 and y =- 74 is a solution. b) If we replace y =- 74 by y = (- 74) + (5 × 37) and x = 37 by x = 37- (2 × 37), then we have (5(37- 2(37))+(2(- 74+(5(37)) = 37 (We have added and subtracted 10 × 37.) or 5(- 37) + 2(111) = 37 . Thus x =- 37 and y = 111. 2. a) Find a polynomial p ( x ) of degree 3 with integral coefficients, such that p ( x ) ≡ 0 (mod 3), for all x ∈ Z . b) Is there a polynomial q ( x ) of degree = 1 or 2 with integral coefficients, such that q ( x ) ≡ 0 (mod 3), for all x ∈ Z ? Explain....
View Full Document

{[ snackBarMessage ]}

Page1 / 3

ex2solsSummer09 - Math 347 C1 HOUR EXAM II 21 July 2009...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online