{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

College Algebra Exam Review 19

# College Algebra Exam Review 19 - 1.6 DIVISIBILITY IN THE...

This preview shows page 1. Sign up to view the full content.

1.6. DIVISIBILITY IN THE INTEGERS 29 existence part of the following assertion. In the following proof, I’m go- ing to arrange the inductive argument in a more streamlined way, but the essential idea is the same. Proposition 1.6.7. Given integers a and d , with d 1 , there exist unique integers q and r such a D qd C r and 0 r < d . Proof. First consider the case a 0 . If a < d , take q D 0 and r D a . Now suppose that a d . Assume inductively that the existence assertion holds for all nonnegative integers that are strictly smaller than a . Then in particular it holds for a d , so there exist integers q 0 and r such that .a d/ D q 0 d C r and 0 r < d . Then a D .q 0 C 1/d C r , and we are done. We deal with the case a < 0 by induction on j a j . If d < a < 0 , take q D 1 and r D a C d . Suppose that a d , and assume inductively that the existence assertion holds for all nonpositive integers whose absolute values are strictly smaller than j a j . Then in particular it holds for a C d , so there exist integers q 0 and r such that .a C d/ D q 0 d C r and 0 r < d .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern