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Disc-a9 - 407 mod 74 = 37 74 mod 37 = 0 so gcd(3478 2183 =...

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1. Find integers q and r so that ° 185 = 22 q + r and 0 ± r < 22 . What are ( ° 185) mod 22 and ( ° 185) div 22 ? One way to do this is to compute ° 185 = 22 = ° 8 : 4091 on a hand calculator. That tells us that q = ° 9 , the integer just to the left of ° 8 : 4091 . So r = ° 185 ° 22 q = ° 185 ° 22( ° 9) = 13 . So ( ° 185) mod 22 = 13 and ( ° 185) div 22 = ° 9 . 2. Use the Euclidean algorithm to °nd gcd (3478 ; 2183) . 3478 mod 2183 = 1295 2183 mod 1295 = 888 1295 mod 888 = 407
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Unformatted text preview: 407 mod 74 = 37 74 mod 37 = 0 , so gcd (3478 ; 2183) = 37 3. Find integers x and y so that 77 x + 144 y = gcd (77 ; 144) . The table for the Euclidean algorithm is 77 x + 144 y x y q 144 1 77 1 1 67 & 1 1 1 10 2 & 1 6 7 & 13 7 1 3 15 & 8 2 1 & 43 23 so gcd (77 : 144) = 1 and 77 ( & 43) + 144 (23) = 1 ....
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