MTH212 Foundations of Elementary Math 2

Fun with Fractions.

Notes on Fractions
Winter 2020

Chapter 4: Number Theory

§ 4.1 Factors and Multiples

Topics

• What is number theory?
The study of patterns in whole numbers. {2,3,5,7,11, ...}
• Factor
4 is a factor of 12.
• Multiple
12 is a multiple of 3.
• Divides:
a|b if a*n=b
a divides b if there is a natural number such that a times n equals b.
• Models:
linear

rectangular
• Use color rods as linear model for factors and multiples:

Color Rods App
• Use color tiles as a rectangular model for factors and multiples:

Color Tiles App
• Even and Odd
• Divisibility Tests
2
the ones digit is a 0,2,4,6,8
3
the sum of the digit is a multiple of 3
4
the two right-most digits form a multiple of 4
5
the ones digit is a 0,5
6
meets the tests for 2 and 3
8
the three right-most digits form a multiple of 8
9
the sum of the digit is a multiple of 9
10
the ones digit is a 0
11
the difference between the sum of the digits from the even powers of 10 and the sum of the digits from the odd powers of 10, is a multiple of 3
• Divisibility Properties:
• If a|b, then a|(n*b),
• If a|b and a|c, then a|(b+c),
• If a|b and a cancel(|)c, then a cancel(|)(b+c)
• Prime and Composite
Play with rectangular arrays
• Factor Trees
• Prime number test
Suppose n is a whole number and k is the smallest whole number such that k × k is greater than n. If there is no prime number less than k that is a factor of n, then n is a prime number.
• Sieve of Eratosthenes