MIT6_042JS10_lec21

MIT6_042JS10_lec21 - 1 Albert R Meyer, March 29, 2010 lec...

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Unformatted text preview: 1 Albert R Meyer, March 29, 2010 lec 8M.1 Intro to Number Theory: Divisibility, GCD’s Mathematics for Computer Science MIT 6.042J/18.062J Albert R Meyer, March 29, 2010 Arithmetic Assumptions assume usual rules for + , ! ,- : a (b+c) = ab + ac, ab = ba, (ab)c = a (bc), a – a =0, a + 0 = a, a+1 > a, …. lec 8M.2 Albert R Meyer, March 29, 2010 The Division Theorem For b > 0 and any a , have q = quotient( a , b ) r = remainder( a , b ) unique numbers q , r such that a = qb + r and 0 r < b . Take this for granted too! lec 8M.3 Albert R Meyer, March 29, 2010 Divisibility c divides a ( c|a ) iff a = k ! c for some k ! 5|15 because 15 = 3 ! 5 n|0 because 0 = 0 ! n lec 8M.4 Albert R Meyer, March 29, 2010 Simple Divisibility Facts • ! c|a implies c|( s a) [ a=k’c implies ( s a)=( s k’)c ] lec 8M.5 k Albert R Meyer, March 29, 2010 Simple Divisibility Facts • ! c|a implies c|( s a) • ! if c|a and c|b then c|(a+b) [if a=k 1 c , b=k 2 c then a+b= (k 1 +k 2 )c ] lec 8M.6 2 Albert R Meyer, March 29, 2010 • ! c|a implies c|(xa) • ! if c|a and c|b then c|(a+b) Simple Divisibility Facts...
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This note was uploaded on 05/27/2011 for the course CS 6.042J taught by Professor Prof.albertr.meyer during the Spring '11 term at MIT.

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MIT6_042JS10_lec21 - 1 Albert R Meyer, March 29, 2010 lec...

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