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

off-topic
May 21, 2019

#1
+2482
+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)

May 26, 2019