Ray's New Elementary Algebra · Arts. 97–102 · Unit 2: Chapter II: Theorems, factoring, etc.

14. Greatest common divisor of monomials and numbers

Goal: Your student finds the G.C.D. of monomials by prime factors, and of two numbers by repeated division.

⏱ About 35 minutes · 7 steps

Open it at the Internet Archive ↗

You'll need

  • The book, open to the page
  • Pencil and plenty of paper (or a whiteboard)
  • A card to cover printed answers

What's in the book

Common divisor and greatest common divisor defined, the rule for monomials with worked examples and exercises 3 to 6, three propositions about measures, and a worked explanation of finding the G.C.D. of 116 and 36 by division.

The lesson, step by step

  1. Warm-up

    Ask: What is the largest number that divides both 8 and 12? (4.)

  2. Definitions

    Read Arts. 97 and 98.

    A Common Divisor, or Common Measure, is any quantity that will exactly divide two or more quantities.

    SayG.C.D. means greatest common divisor. It is the biggest thing that goes into all of them.

  3. The rule for monomials

    Read Art. 100 and its rule:

    1. Resolve the quantities into their prime factors.

    2. Take the product of those factors that are common to each of the terms for the greatest common divisor.

    Work the book's first example: 6ab = 2 × 3ab and 15a²c = 3 × 5aac; the common factors are 3 and a, so the G.C.D. is 3a.

  4. Guided practice

    Work example 2 together (4a²x³, 6a³x² and 10a⁴x, G.C.D. 2a²x). Point out the note: for letters, take the lowest power that appears in all. Then your student does exercises 3 to 6, covering the answers.

  5. The three propositions

    Read Art. 101. Test each with numbers: 3 measures 6, so it measures 12 and 18; 3 measures 27 and 18, so it measures 45 and 9.

  6. Division method for numbers

    Read Art. 102 and work it on paper: 116 ÷ 36 = 3 remainder 8; 36 ÷ 8 = 4 remainder 4; 8 ÷ 4 = 2 remainder 0. The G.C.D. is 4. Your student then finds the G.C.D. of 84 and 120 the same way. (12.)

  7. Wrap-up

    Ask: Why must the G.C.D. of 116 and 36 also divide the remainder 8? (Proposition III: it measures their difference.)

Tip

This is a shorter lesson. If time is left, review factoring trinomials from last session.

Answers to the book's problems

The book prints the answers to exercises 3 to 6. For the extra practice in the guide: the G.C.D. of 84 and 120 is 12.

Worked out for this site from the scan. If a number in your copy differs, trust the book. Every answer is in the Key ↗.

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