# Integer solution for $n_1 k_1 + n_2 k_2 + n_3 k_3 = 1$

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

• Have you heard of Bezout's Lemma? – Ishan Banerjee Feb 22 '13 at 10:35
• No I have not. Sorry – richard Feb 22 '13 at 10:38

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 $$\gcd(n_1,n_2,n_3)=1$$