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By Ghiani G., Laporte G.

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D + 1) and obtain the parametric residues r(xlb X2b ... , X(n-l)b xn) in Xn. 4 for each one variable in succession, until we totally reconstruct r(xl> ... , xn). In numerical analysis, such an approximation or interpolation is known as product operator method (see Schumaker [1977], McKinney [1972], Hartley [1976]). We will see more about this in the next chapter. EXERCISES I 1. Find the gcd(601,101). 2. Find the gcd(3125,195). 3. Find the icm(25,15). 4. Find the mod 125 inverse of 17. 5. Find the mod 19 inverse of 11.

The remaining steps to prove the validity are identical to the case of OM, and we obtain Rn-l(x) = reX). 3 EXAMPLE. (i) Let Po= 13, PI = 11, P2=7 r 1 = 2, r2=4. ro=4, The algorithmic steps are shown below: Pk M IL d 4 13 2 11 7 1 13 143 6 5 10 1 k 'k 0 1 2 4 Rk 4 134 277 Thus R2 = r = 277 . (ii) Let Po(x)=x-1, ro = r(xo) = 3, over (0 7 , +, '). 5 Residue representation of signed integers We now briefly recall how we represent both positive and negative integers in the residue or modular representation (see Chapter I, Gregory and Krishnamurthy [1984]).

1 Definition. 1·lm:~ ~ §m S E §m by the is defined by writing Ibl m == s(mod m) if and only if b == s(mod m) and -m m -

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A Branch-and-cut Algorithm for the Undirected Rural Postman Problem by Ghiani G., Laporte G.

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