Ray's New Elementary Algebra · Arts. 107–112 · Unit 2: Chapter II: Theorems, factoring, etc.

16. Least common multiple

Goal: Your student finds the L.C.M. of monomials and polynomials, by the division rule or by prime factors.

⏱ 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

Multiple, common multiple and least common multiple defined, why the L.C.M. holds each prime factor once, the rule with a worked example (ax, bx and abc), the shortcut product ÷ G.C.D., and five exercises with printed answers. A general review of the chapter follows.

The lesson, step by step

  1. Warm-up

    Ask: What is the smallest number that both 4 and 6 go into? (12.)

  2. Definitions

    Read Arts. 107 to 109.

    The Least Common Multiple of two or more quantities is the least quantity that will contain them exactly.

    SayG.C.D. is the biggest thing that goes INTO them; L.C.M. is the smallest thing they all go into.

  3. Why it works

    Read Arts. 110 and 111. Say: 30 does not contain 14 because 14 has the factor 7 and 30 does not. The L.C.M. must have every prime factor, but no extra ones.

  4. The rule, worked together

    Read the rule and work the book's example with ax, bx and abc in a line: divide by a, then x, then b. The divisors a, x, b and what is left, c, multiply to abcx.

    1. Arrange the quantities in a horizontal line.

    Then show the other way: write each in prime factors and take each factor the greatest number of times it appears.

  5. Guided practice

    Cover the answers. Your student does exercises 1 and 2 by prime factors, then exercise 4, 4a²(a − x) and 6ax²(a² − x²), noticing that a² − x² already contains a − x.

  6. The shortcut

    Read Art. 112: the product of two quantities divided by their G.C.D. is their L.C.M. Test it: 8 × 12 = 96, G.C.D. 4, and 96 ÷ 4 = 24.

  7. Wrap-up

    Use the General Review questions at the end of the chapter as an oral quiz. Pick eight or ten.

Tip

The General Review on p. 64 is a good end-of-chapter check. Note any weak spots for review before fractions.

Answers to the book's problems

The book prints the answer after each exercise.

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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