Ray's New Higher Arithmetic · Art. 91 · Unit 7: Properties of Numbers
24. Kinds of numbers
Goal: Your student classifies whole numbers as prime or composite, even or odd, perfect or imperfect, and uses the words divisor, multiple, factor and aliquot part.
You'll need
- The book
- A slate, small whiteboard or scrap paper
- A notebook and pencil
What's in the book
Article 91's seventeen numbered definitions: integers, prime and composite numbers, numbers prime to each other, even and odd, perfect and imperfect (abundant and defective), divisor, multiple, factor, prime factor and aliquot part.
The lesson, step by step
Warm-up
Ask: Which numbers from 1 to 20 can be split into equal rows (more than one row, more than one in a row)? Lay out beans if that helps.
Prime and composite
Read:
A prime number is one that can be exactly divided by no other whole number but itself and unity.
A composite number is one that can be exactly divided by some other whole number besides itself and unity.
Note: the book counts 1 as prime. Most modern books don't; mention it, but don't fuss.
Prime to each other
Read:
Two numbers are prime to each other, when unity is the only number that will exactly divide both.
The book's example 4 and 9: both composite, but prime to each other.
Even, odd, perfect
Read:
An even number is one which can be divided by 2 without a remainder.
A perfect number equals the sum of its divisors: 6 = 1 + 2 + 3, and 28 = 1 + 2 + 4 + 7 + 14. 18 is abundant (1+2+3+6+9 = 21), and 16 is defective (1+2+4+8 = 15).
Divisors, multiples and parts
Read:
Any factor of a number is called an aliquot part of it.
List the aliquot parts of 12 with the book: 1, 2, 3, 4, 6.
Practice
Your student lists the primes up to 50 and classifies 24, 25, 28, 29 and 30 using every word from today.
Wrap-up
Ask: Is every odd number prime? (No: 9, 15, 21...)
Answers to the book's problems
Primes to 50 (the modern list, without 1): 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. 24: composite, even, abundant. 25: composite, odd, defective. 28: composite, even, perfect. 29: prime, odd. 30: composite, even, abundant.
Worked out for this site from the scan. If a number in your copy differs, trust the book.