Prove the theorem (elementary number theory) * 1.51 Theorem. Let a, b. c, Xo. and yo be

Prove the theorem (elementary number theory)

 

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* 1.51 Theorem. Let a, b. c, Xo. and yo be integers with a and b not both = c. Then the integers 0 such that axo +byo a X = xo + and (a, b) y = Vo (a, b) also satisfy the linear Diophantine equation ax + by = c. This theorem leaves open the question of whether this method of gener- ating alternative solutions generates all the solutions or whether there are yet more solutions. * 1.51 Theorem. Let a, b. c, Xo. and yo be integers with a and b not both = c. Then the integers 0 such that axo +byo a X = xo + and (a, b) y = Vo (a, b) also satisfy the linear Diophantine equation ax + by = c. This theorem leaves open the question of whether this method of gener- ating alternative solutions generates all the solutions or whether there are yet more solutions.

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