Ray's New Higher Arithmetic · Arts. 92–94 · Unit 7: Properties of Numbers
25. Factoring and tests of divisibility
Goal: Your student tests a number for divisibility by 2, 3, 4, 5, 6, 9 and 10 at a glance and breaks any number into prime factors.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
Factoring: two principles, seven propositions on divisibility (by 2, 4, 5, 10, composite products, 6, primes ending in 1, 3, 7 or 9, and 9 and 3 by the sum of digits), and the rule for finding prime factors with 11 examples (answers printed).
The lesson, step by step
Warm-up
Ask: Is 3,456 divisible by 2? By 3? How did you decide?
Principles
Read:
A factor of a number is a factor of any multiple of that number.
A factor of any two numbers is also a factor of their sum.
These two explain every divisibility test that follows.
The tests
Read the propositions:
Every number ending with 0, 2, 4, 6, or 8, is divisible by 2.
A number is divisible by 4, when the number denoted by its two right-hand digits is divisible by 4.
A number ending in 0 or 5 is divisible by 5.
If any even number is divisible by 3, it is also divisible by 6.
And Prop. VII: a number is divisible by 9 (or 3) when the sum of its digits is.
Why the 4-test works
The book's reason: 100 is divisible by 4, so all the hundreds are. Only the last two digits can matter. Try 384: 84 ÷ 4 = 21, so yes.
Prime factors
Read:
Divide the given number by any prime number that will exactly divide it
and keep going until the quotient is prime. Work 42 = 2 × 3 × 7 with the book, then the first examples:
45. Ans. 3, 3, 5.
54. Ans. 2, 3, 3, 3.
Independent practice
Your student finds the prime factors of all 11 examples. The book prints the answers.
Wrap-up
Ask: Why is the least divisor of any number (other than 1) always prime? (If it were composite, its own factors would be smaller divisors.)