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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)?

 Jun 9, 2021
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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)?

 

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

 

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

 

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

 

laugh

 Jun 9, 2021

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