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week07 Euc

# week07 Euc - 1 then r 2 ∴ it divides r k-1 too Working...

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The Euclidean Algorithm Let d = greatest common divisor of a + b. Find d as follows: (Suppose b < a; reverse if otherwise ) a = bq + r b = r q 1 + r 1 r = r 1 q 2 + r 2 r 1 = r 2 q 3 + r 3 .................. r k-2 = r k-1 q k + r k r k-1 = r k q k+1 Then r k is greatest common divisor d. For, if any number divides a and b, it divides r, then divides b and r, must divide r
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Unformatted text preview: 1 , then r 2 , . .. ∴ it divides r k-1 too. Working backwards, r k divides r k-1 , thus r k-2 , up to a + b. ∴ r k is g. c . d. Note also : r k is a linear combination of a and b. © P. Rosenthal , University of Toronto, Department of Mathematics typed by A. Ku Ong...
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