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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.

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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 .

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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 )

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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...

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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 . ...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 Can compute with representative in { 0 ,..., m - 1 } .

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Modular Arithmetic: refresher.
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