We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.

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

 May 21, 2019

Let's convert the equation into base 10



\(AAA_7\) = \(56A+1\)


\(BBB_7\) = \(56B+1\)


And \(666_7\) = \(337\)




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

9 Online Users