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For given integers $k_1,k_2,k_3$ is there an integer solution for the following equation: $$n_1 k_1 + n_2 k_2 + n_3 k_3 = 1$$

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  • $\begingroup$ Have you heard of Bezout's Lemma? $\endgroup$ – Ishan Banerjee Feb 22 '13 at 10:35
  • $\begingroup$ No I have not. Sorry $\endgroup$ – richard Feb 22 '13 at 10:38
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Bezout's Identity says that for any given integers $n_1$ and $n_2$ there are integers $k_1$ and $k_2$ so that $$ k_1n_1+k_2n_2=\gcd(n_1,n_2) $$ Simply extending this, we get that for any given integers $n_1$, $n_2$, and $n_3$ there are integers $k_1$, $k_2$, and $k_3$ so that $$ k_1n_1+k_2n_2+k_3n_3=\gcd(n_1,n_2,n_3) $$ Thus, there is an integer solution for $$ k_1n_1+k_2n_2+k_3n_3=1 $$ if and only if $$ \gcd(n_1,n_2,n_3)=1 $$

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