We have 2 numbers: AB base 8 and BA base 7, where A and B are variables. If both of the numbers they have are equal, what is A + B?
I have attempted this problem, trying to convert both to base 10 and use those to compare the two, but I don't get an answer. Can someone help?
I keep getting 67 and 76, which would work only if 76 was a real number in base 7 .
$$8A+B=7B+A\implies 7A=6B$$ The integral solutions of $7A=6B$ are $A=6t,B=7t,t\ge0$. However, writing $BA_7$ implies that $A,B\in[0,7)$ and there is only one $t$ that makes $A,B$ fall in this range: $t=0$. Thus $A=B=A+B=0$.