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# Number Theory

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The six-digit integer 978,XYZ consists of six distinct digits and is divisible by 7, 8 and 9.  What is the three-digit integer XYZ?

May 25, 2021

#1
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XYZ = 420.

May 26, 2021
#2
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The six-digit integer 978XYZ consists of six distinct digits and is divisible by

7, 8 and 9.
What is the three-digit integer XYZ?

$$\begin{array}{|rcll|} \hline 7 | 978XYZ \\ 8 | 978XYZ \\ 9 | 978XYZ \\ \hline \end{array} \begin{array}{|rcll|} \hline XYZ =~ ? \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline 7*8*9 &=& 504 \\ 978000 : 504 &=& 1940~ R~ 240 \\ 240+\mathbf{264} &=& 504 \\ \mathbf{XYZ} &=& \mathbf{264} \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline 978000 +264 &=& 978264 \\ 978264 : 7 &=& 139752 \\ 978264 : 8 &=& 122283 \\ 978264 : 9 &=& 108696 \\ \hline \end{array}$$

May 26, 2021