1. Modular Arithmetic and Change of Base

2. Modular Arithmetic
Let a and b be integers and let m be a positive integer. Then if and only if a and b have the same integer remainder when they are divided by m.
Congruence Theorem: integers a and b and positive integers m, if and only if m is a factor of a - b

3. Example 1
Show that using the first definition and the Congruence Theorem.

4. Example 2
Find three numbers for each of the congruencies.

5. Properties of Modular Arithmetic
Let a, b, c, and d be any integers and let m be a positive integer. If and then:
Addition Property of Congruence: _____________________________________
Subtraction Property of Congruence: ____________________________________
Multiplication Property of Congruence: _____________________________________

6. Example 3
Find the last three digits of 199.

7. Number Bases and Conversions
Base 2 or binary notation: numbers represented by 0 and 1
Other bases uses similar numbers for example base 3 uses 0, 1, and 2 and base 7 uses 0, 1, 2, 3, 4, 5, and 6.

8. Example 4 and 5
Give the base 10 representation for the number
Write 289 in base 2.

9. Example 6 and 7
Change to base 10.
Change 78 to base 3.