Existence of Integer solution for a set of linear equations

Question

Can anyone give a proof sketch of the following claim: If a system of homogeneous linear equations with integer coefficients has a positive real solution, then it also has a positive integer solution?

Thanks in advance!

Answer 1

Think about how you solve systems of linear equations, say Gaussian elimination. Note that at each step you are just dividing, multiplying, subtracting, or adding coefficients. The rationals are closed under these operations, so if you start with rational coefficients (which integers are) then your solution (if it exists) will also have only rational values for the unknown. Write them as a fractions $x_1 = p_1/q_1, ..., x_n = p_n/q_n$ where $\text{gcd}(p_i,q_i) = 1$ and let $k = q_1\cdot q_2\cdot...\cdot q_n$.

If your original system was $Ax = 0$, will the solutions to $kAx = 0$ be any different? Now look at $A(kx) = 0$.