Modular Arithmetic and Change of Base

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Presentation transcript:

Modular Arithmetic and Change of Base

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

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

Example 2 Find three numbers for each of the congruencies.

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: _____________________________________

Example 3 Find the last three digits of 199.

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.

Example 4 and 5 Give the base 10 representation for the number 110110112. Write 289 in base 2.

Example 6 and 7 Change 21103 to base 10. Change 78 to base 3.