The Question: We can define a set of integers $X_a$$_,$$_b$ = {∀u, v ∈ $\Bbb Z$, au + bv}. For example, if a = 6 and b = 8 then X$_6$$_,$$_8$ includes numbers like 20 = 2$*$6 + 1$*$8 and 4 = −2$*$6 + 2$*$8. Let c be the smallest positive integer in X$_a$$_,$$_b$. Prove that every number in X$_a$$_,$$_b$ is a multiple of c.
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Have you tried by contradiction? If $c$ doesn't divide $m$, then there is a non-zero remainder... – Arnaud Mortier Oct 06 '18 at 22:36
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@ArnaudMortier How does a non-zero remainder have any connection to X$_{a,b}$? – Cup Oct 07 '18 at 00:00
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Divide the arbitrary element of $X_{a,b}$ by $c$
The reminder if not zero is a positive number less than $c$ which is also an element of $X_{a,b}$
I let you show that the remainder is indeed an element of $X_{a,b}.$
Mohammad Riazi-Kermani
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How to prove the remainder is an element of X${a,b}$? Assume X${a,b}$ = k$1$a + k$_2$b, since k1 and k2 can be anything, does that mean r is an element of X${a,b}$ – Cup Oct 06 '18 at 22:47
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The remainder is the difference of two elements so it is an element. Think about linear combinations. – Mohammad Riazi-Kermani Oct 06 '18 at 22:54
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Im sorry but i still dont really understand. What would the two elements be? – Cup Oct 07 '18 at 00:06