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For how many ordered pairs $(A,B)$ where $A$ and $B$ are positive integers is $AAA_7+BBB_7=666_7?$

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 May 21, 2019
 #1
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Let's convert the equation into base 10

 

 

\(AAA_7\) = \(56A+1\)

 

\(BBB_7\) = \(56B+1\)

 

And \(666_7\) = \(337\)

 

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So we have : \(56A+56B=335\)

 

OK.... Now we need to find how many (A,B) there is.

 

To find the maximum and minimum one of the variables can be (A or B) we do \(\lfloor\frac{335}{56}\rfloor\)

 

So maximum can be 5

 

The minimum is obviously 1.

 

So we have to substitute 1, 2, 3, 4, 5, for A or B to see if there are solutions.

 

There are no integer solutions.

 

So there are zero ordered pairs (A,B)

 May 26, 2019

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