ยง1.1 Exercises

Exercises 1.1.A

Exercise 1.1.1

In each part, find the quotient \(q\) and remainder \(r\) when \(a\) is divided by \(b\), without using technology. Check your answers.

  • (a) \(a = 17; \ b = 4\)
  • (b) \(a = 0; \ b = 19\)
  • (c) \(a = -17; \ b = 4\)

Exercise 1.1.2

  • (a) \(a = -51; \ b = 6\)
  • (b) \(a = 302; \ b = 19\)
  • (c) \(a = 2000; \ b = 17\)

Exercise 1.1.3

Use a calculator to find the quotient \(q\) and remainder \(r\) when \(a\) is divided by \(b\).

  • (a) \(a = 517; \ b = 83\)
  • (b) \(a = -612; \ b = 74\)
  • (c) \(a = 7,965,532; \ b = 127\)

Exercise 1.1.4

  • (a) \(a = 8,126,493; \ b = 541\)
  • (b) \(a = -9,217,645; \ b = 617\)
  • (c) \(a = 171,819,920; \ b = 4321\)

Exercise 1.1.5

Let \(a\) be any integer and let \(b\) and \(c\) be positive integers. Suppose that when \(a\) is divided by \(b\), the quotient is \(q\) and the remainder is \(r\), so that:

\(a = bq + r \quad \text{and} \quad 0 \leq r < b.\)

If \(ac\) is divided by \(bc\), show that the quotient is \(q\) and the remainder is \(rc\).

Exercises 1.1.B

Exercise 1.1.6

Let \(a\), \(b\), \(c\), and \(q\) be as in Exercise 1.1.5. Suppose that when \(q\) is divided by \(c\), the quotient is \(k\). Prove that when \(a\) is divided by \(bc\), the quotient is also \(k\).

Exercise 1.1.7

Prove that the square of any integer \(a\) is either of the form \(3k\) or of the form \(3k + 1\) for some integer \(k\).

Hint: By the Division Algorithm, \(a\) must be of the form \(3q\), \(3q + 1\), or \(3q + 2\).

Exercise 1.1.8

Use the Division Algorithm to prove that every odd integer is either of the form \(4k + 1\) or of the form \(4k + 3\) for some integer \(k\).

Exercise 1.1.9

Prove that the cube of any integer \(a\) must be exactly one of these forms: \(9k\), \(9k + 1\), or \(9k + 8\) for some integer \(k\).

Hint: Adapt the hint in Exercise 1.1.7, and cube \(a\) in each case.

Exercise 1.1.10

Let \(n\) be a positive integer. Prove that \(a\) and \(c\) leave the same remainder when divided by \(n\) if and only if \(a - c = nk\) for some integer \(k\).

Exercise 1.1.11

Prove the following version of the Division Algorithm, which holds for both positive and negative divisors.

Extended Division Algorithm: Let \(a\) and \(b\) be integers with \(b \neq 0\). Then there exist unique integers \(q\) and \(r\) such that \(a = bq + r\) and \(0 \leq r < |b|.\)

Hint: Apply Theorem 1.1 when \(a\) is divided by \(|b|\). Then consider two cases \((b > 0\) and \(b < 0)\).