Positive integers $k_1,k_2,...,k_n,K$ are given where $K$ is divisible by each of the numbers $k_1,k_2,...,k_n$. Suppose there are integers $x_1,x_2,...,x_n$ which satisfy the following equation $$\frac{K}{k_1}\cdot x_1+\frac{K}{k_2}\cdot x_2+...+\frac{K}{k_n}\cdot x_n=1$$ Prove that there are also such integers $y_1,y_2,...,y_n$ that $$\frac{K}{k_1}\cdot y_1+\frac{K}{k_2}\cdot y_2+...+\frac{K}{k_n}\cdot y_n=1$$ and $|y_i|\leqslant k_i$ for all $i=1,2,...,n$
I tried to look for some sort of an algorithm here because I think there is no other way to solve it. Please let me know if you know how I should approach this problem. I am always grateful for your help
Hint: Add 0 creatively by adding 1 to a term while subtracting 1 from another term.
Further Hint: If there exists a $x_I > k_I$, then there exists a $x_J < 0$. Replace $x_I \rightarrow x_I - k_I, x_J \rightarrow x_J + k_J$.
Show that we can do this finitely often till all $x_ i \leq k_i$ and likewise $-k_i \leq x_i$.