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Unformatted text preview: Administration Please, be considerate in that last minute!! Administration Please, be considerate in that last minute!! Thanks!! CS70: Satish Rao: Lecture 8. Outline. 1. Modular Arithmetic: review. 2. Inverses for Modular Arithmetc: Greatest Common Divisor. 3. Euclid’s GCD Algorithm 4. Euclid’s Extended GCD Algorithm. Modular Arithmetic: refresher. x is congruent to y modulo m or “ x ≡ y ( mod m ) ” if ( x y ) is divisible by m . Modular Arithmetic: refresher. x is congruent to y modulo m or “ x ≡ y ( mod m ) ” if ( x y ) is divisible by m . ...or x and y have the same remainder w.r.t. m . Modular Arithmetic: refresher. x is congruent to y modulo m or “ x ≡ y ( mod m ) ” if ( x y ) is divisible by m . ...or x and y have the same remainder w.r.t. m . Fact: Addition, subtraction, multiplication can be done with any equivalent x and y . Modular Arithmetic: refresher. x is congruent to y modulo m or “ x ≡ y ( mod m ) ” if ( x y ) is divisible by m . ...or x and y have the same remainder w.r.t. m . Fact: Addition, subtraction, multiplication can be done with any equivalent x and y . or “ a ≡ c ( mod m ) and b ≡ d ( mod m ) = ⇒ a + b ≡ c + d ( mod m ) and a · b = c · d ( mod m ) ” Modular Arithmetic: refresher. x is congruent to y modulo m or “ x ≡ y ( mod m ) ” if ( x y ) is divisible by m . ...or x and y have the same remainder w.r.t. m . Fact: Addition, subtraction, multiplication can be done with any equivalent x and y . or “ a ≡ c ( mod m ) and b ≡ d ( mod m ) = ⇒ a + b ≡ c + d ( mod m ) and a · b = c · d ( mod m ) ” Proof Idea: Do addition(multiplication) in regular arithmetic results differ by multiple of m . Modular Arithmetic: refresher. x is congruent to y modulo m or “ x ≡ y ( mod m ) ” if ( x y ) is divisible by m . ...or x and y have the same remainder w.r.t. m . Fact: Addition, subtraction, multiplication can be done with any equivalent x and y . or “ a ≡ c ( mod m ) and b ≡ d ( mod m ) = ⇒ a + b ≡ c + d ( mod m ) and a · b = c · d ( mod m ) ” Proof Idea: Do addition(multiplication) in regular arithmetic results differ by multiple of m . QED... Modular Arithmetic: refresher. x is congruent to y modulo m or “ x ≡ y ( mod m ) ” if ( x y ) is divisible by m . ...or x and y have the same remainder w.r.t. m . Fact: Addition, subtraction, multiplication can be done with any equivalent x and y . or “ a ≡ c ( mod m ) and b ≡ d ( mod m ) = ⇒ a + b ≡ c + d ( mod m ) and a · b = c · d ( mod m ) ” Proof Idea: Do addition(multiplication) in regular arithmetic results differ by multiple of m . QED... ish Modular Arithmetic: refresher. x is congruent to y modulo m or “ x ≡ y ( mod m ) ” if ( x y ) is divisible by m ....
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This note was uploaded on 02/29/2012 for the course COMPSCI 70 taught by Professor Rau during the Fall '11 term at Berkeley.
 Fall '11
 Rau

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