{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecnotes2

# lecnotes2 - Lecture Notes Concepts of Mathematics(21-127...

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

Lecture Notes, Concepts of Mathematics (21-127) Lecture 1, Recitation A–D, Spring 2008 2 Integers and Diophantine Equations 2.1 The Division Algorithm READ: Textbook pages 25–34. EXERCISES: Pages 50–55, #1–30, 73, 74 Recall the definition of divisibility: If a and b are integers, then a divides b iff there is an integer q such that aq = b . Proposition 2.1 (Proposition 2.11, page 25) Let a , b , and c be integers. (i) If a | b and b | c , then a | c . (ii) If a | b and a | c , then a | ( bx + cy ) for any x, y Z . In particular, a | ( b + c ) and a | ( b - c ) . (iii) If a | b and b | a , then a = ± b . (iv) If a | b and b = 0 , then | a | ≤ | b | . Proof: (ii) Assume a | b , a | c , and let x, y Z . Since a | b and a | c , there are q 1 , q 2 Z such that aq 1 = b and aq 2 = c . Then we have aq 1 x = bx and aq 2 y = cy , and thus bx + cy = aq 1 x + aq 2 y = a ( q 1 x + q 2 y ) . Since q 1 x + q 2 y Z , we have that a | ( bx + cy ). (iv) Assume a | b and b = 0. Then there is a q Z such that aq = b . Since b = 0, this implies q = 0. Since q is an integer, | q | ≥ 1, so | a | ≤ | a | · | q | = | aq | = | b | .

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

lecnotes2 - Lecture Notes Concepts of Mathematics(21-127...

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

View Full Document
Ask a homework question - tutors are online