Number Theory

The letters A, B, C, D, E, and F represent digits and 

ABC,DEF represents a positive six-digit integer.

What is the number ABC,DEF if 4(ABC,DEF) = 9(DEF,ABC)?

$\text{Let $ABC = x$ and $DEF = y$ }$

$\begin{array}{|rcll|} \hline 4(ABC,DEF) &=& 4(1000x+y) \\\\ 9(DEF,ABC) &=& 9(1000y+x) \\\\ \hline \\ 4(1000x+y) &=& 9(1000y+x) \\\\ 4000x+4y &=& 9000y+9x \\\\ 3991x &=& 8996y \\\\ 13*307x &=& 13*692y \\\\ 307x &=& 692y \\ \hline \mathbf{ x } &=& \mathbf{692} \\\\ \mathbf{ y } &=& \mathbf{307} \\ \hline \end{array}$

$4(692,307) = 9(307,692)$