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LDEExtraExample

LDEExtraExample - n ≥-90 or n ≥-2 22 34 or n...

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MATH 135 Fall 2005 LDE Example Find all non-negative integer solutions to 174 x - 68 y = 20 with 2 y - 5 x 10. Solution First, we set w = - y to obtain the LDE 174 x + 68 w = 20. Using the EEA, we find that gcd(174 , 68) = 2 and 174(9) + 68( - 23) = 2. Thus, since 2 | 20, then 174 x + 68 w = 20 has a solution. In particular, 174(90) + 68( - 230) = 20. Therefore, the complete solution to 174 x + 68 w = 20 is x = 90 + n 68 2 = 90 + 34 n w = - 230 - n 174 2 = - 230 - 87 n for all n Z Therefore, the complete solution to 174 x - 68 y = 20 is x = 90 + 34 n y = 230 + 87 n for all n Z since y = - w . Since we are looking for non-negative solutions, then (i) x 0 so 90 + 34 n
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Unformatted text preview: n ≥ -90 or n ≥ -2 22 34 or n ≥ -2 (since n ∈ Z ), and (ii) y ≥ 0 so 230 + 87 n ≥ 0 or 87 n ≥ -230 or n ≥ -2 56 87 or n ≥ -2 (since n ∈ Z ) Also, we want 2 y-5 x ≤ 10, so 2(230 + 87 n )-5(90 + 34 n ) ≤ 10 or 10 + 4 n ≤ 10 or n ≤ 0. Therefore, our three conditions are n ≥ -2 and n ≥ -2 and n ≤ 0, which give-2 ≤ n ≤ 0. Therefore, the solutions which satisfy the conditions correspond to n =-2, n =-1 and n = 0, namely ( x, y ) = (90 , 230) , (56 , 143) , (22 , 56)....
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