CSC-7: Chapter 4: Section 4: Division into Cases and the Quotient-Remainder Theorem

(4) According to the quotient-remainder theorem, if an integer n is divided by a positive integer d, the possible remainders are _____. This implies that n can be written in one of the forms _____ for some integer q.

0, 1, 2, ..., (d-1); dq, dq + 1, dq + 2, ..., dq + (d - 1)

The square of any odd integer has the form _________________. Theorem 4.4.3

8m + 1 for some integer m. Case 1: (n = 4q + 1, for some integer q) Case 2: (n = 4q + 3 for some integer q)

Theorem The Triangle Inequality

Case 1:

Parity

Commonly said "the parity of an integer" which refers if the integer is even or odd.

(5) To prove a statement of the form "If A1 or A2 or A3, then C," prove _____ and _____ and _____.

If A_1, then C; If A_2, then C; If A_3, then C

Method of Proof by Division into Cases

Similar to the if-else-then statement in computer programming. I would argue it feels like a switch statement too.

How do us programmers see Quotient-Remainder Theorem in everyday life?

We see this with functions such as div (division) and mod (modulus operator, gets the remainder).

Parity property

Whether an integer is even or odd

Complete the phrase The div and mod functions give the values that satisfy the Quotient-Remainder Theorem so long as ______________________.

a nonnegative integer n is divided by a positive integer d.

Lemma

a statement that does not have much intrinsic interest but is helpful in deriving other results

(1) The quotient-remainder theorem says that for all integers n and d with d≥0, there exist _____ q and r such that _____ and _____.

integers n = dq + r; 0 ≤ r ≤ 2

(3) The parity of an integer indicates whether the integer is _____.

odd or even.

Two integers are called consecutive iff

one is one more than the other. Ex Assume m is an integer. m, the consecutive integer is m+1

Triangle inequality

says that the absolute value of two numbers is less than or equal to the sum of their absolute values.

(2) If n and d are integers with d > 0, n div d is _____ and n mod d is _____.

the quotient obtained when n is divided by d; the nonnegative remainder obtained when n is divided by d.