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

Guest May 21, 2019

#1**+1 **

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)

CalculatorUser May 26, 2019