For how many ordered pairs $(A,B)$ where $A$ and $B$ are positive integers is $AAA_7+BBB_7=666_7?$
+2864 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)
