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Summer Clouds moderators

06/11/2017 at 15:33
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The units digit of a positive three-digit integer is 0. The sum of the other two digits is 12. Interchanging the tens and hundreds digits increases the number by 540. What is the original number?




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    Dao Trong Luan Coordinator 06/11/2017 at 17:01

    Tell the number to find is \(\overline{ab0}\)

    \(\Rightarrow\left\{{}\begin{matrix}a+b=12\\\overline{ba0}-\overline{ab0}=540\end{matrix}\right.;a< b\)because \(\overline{ba0}>\overline{ab0}\)

    => \(\left(100b+10a\right)-\left(100a+10b\right)=540\)

    => 100b + 10a - 100a - 10b = 540

    => 90b - 90a = 540

    \(\Rightarrow90\left(b-a\right)=540\)

    \(\Rightarrow b-a=6\)

    \(So-\left\{{}\begin{matrix}a=\dfrac{12-6}{2}=3\\b=\dfrac{12+6}{2}=9\end{matrix}\right.\)

    So the original number is 390

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