Ray's New Higher Arithmetic · Arts. 95–100 · Unit 7: Properties of Numbers
26. Common factors, all divisors and the greatest common divisor
Goal: Your student finds the common prime factors of numbers, lists all divisors of a number, and finds the greatest common divisor by factoring and by successive division.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
Finding prime factors common to several numbers (Art. 95) and all divisors of a number (Art. 96), each with four examples; then the greatest common divisor: definitions, seven principles, Case I by factoring and Case II by successive division, with the rule and examples (answers printed, except one).
The lesson, step by step
Warm-up
Mental: What is the biggest number that divides both 12 and 18?
Common factors and divisors
Read Art. 96's rule:
Resolve the number into its prime factors, and then form from these factors all the different products
Work 42 = 2 × 3 × 7: divisors 2, 3, 7, 6, 14, 21 (and 1 and 42). Then do Art. 95's first two together:
1. 42 and 98. Ans. 2, 7.
2. 45 and 105. Ans. 3, 5.
The idea of the G. C. D.
Read:
A common divisor (C. D.) of two or more numbers, is a number that exactly divides each of them.
The product of all the prime factors, common to two or more numbers, is their greatest common divisor.
Work the book's example: 30 = 2 × 3 × 5 and 105 = 3 × 5 × 7, so the G. C. D. is 3 × 5 = 15.
Successive division
When numbers are hard to factor, divide. Read:
Divide the greater number by the less, and the divisor by the remainder
the last divisor will be the greatest common divisor sought.
Work the book's example 348 and 1024: remainders 328, 20, 8, 4, 0. G. C. D. = 4.
Guided practice
Together:
1. 85 and 120. Ans. 5.
2. 91 and 133. Ans. 1.
What does a G. C. D. of 1 mean? (The numbers are prime to each other.)
Independent practice
Your student works the rest of the examples under both cases, and Art. 96's Example 4 on 496 (answer below).
Wrap-up
Ask: To find the G. C. D. of three numbers, what do you do? (Find it for two, then for that result and the third.)
Afterwards
If time runs short, do Case II (successive division) on a second day.
Answers to the book's problems
Art. 96, Example 4: the divisors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248 (and 496). 496 = 2 × 2 × 2 × 2 × 31; it is even and composite, and it is a perfect number, since 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496.
Worked out for this site from the scan. If a number in your copy differs, trust the book.