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A linear combination, $ax + by +...=z$ where $a, b,..., z \in \mathbb{Z}$. It is bound to yield an integer by the closure property of integer under the addition operation. This fact is used in computing gcd among others.

I want to know about properties of non-integer combination, i.e. given $a, b, ...\in \mathbb {Z}$; but the multipliers $x, y, .. $ not all $\in \mathbb{Z}$. I hope that they must be enjoying similar properties, as they are comprised of rationals, and the rationals are closed under addition too.

If so, then how these properties can be used where the linear combinations do not hold.

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  • $\begingroup$ Can you give a specific example regarding to your question? $\endgroup$ – Berci Nov 26 '17 at 12:37
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    $\begingroup$ In gcd computation, the invariant property (gcd is same at each step) is a product of two linear equations being followed at each step that lead to common divisors of two pairs: (i) remainder ($r$), divisor($a$), & (ii) dividend ($d$), and divisor($a$). In (i) & (ii), $d$ & $r$ are the linear combinations respectively. Quotient ($q$) and divisor ($a$) are not linear combinations as when taken on lhs lead to rhs side expressions of $\dfrac{d-r}{a}$ & $\dfrac{d-r}{q}$ respectively. I want to know, as curiosity, what properties are enjoyed by the two quantities that are not linear combination. $\endgroup$ – jiten Nov 26 '17 at 12:51

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