1. **Problem statement:** We have two numbers: $(ab)_4$ in base 4 and $(ba)_7$ in base 7, where $a$ and $b$ are digits. We want to find the largest possible value of $(a+b)_{10}$ and determine if there exist non-zero digits $a$ and $b$ such that $(ab)_4 = (ba)_7$. 2. **Understanding the problem:** - $(ab)_4$ means the number in base 4 with digits $a$ and $b$, so its decimal value is $4a + b$. - $(ba)_7$ means the number in base 7 with digits $b$ and $a$, so its decimal value is $7b + a$. 3. **Set up the equation:** $$4a + b = 7b + a$$ 4. **Simplify the equation:** $$4a + b = 7b + a$$ $$4a - a = 7b - b$$ $$3a = 6b$$ $$a = 2b$$ 5. **Constraints:** - Since $a$ and $b$ are digits in base 4 and base 7 respectively, $a$ must be in $\{0,1,2,3\}$ and $b$ in $\{0,1,2,3,4,5,6\}$. - Both $a$ and $b$ are non-zero. - From $a=2b$, $a$ must be at most 3, so $2b \leq 3 \Rightarrow b \leq 1.5$. - Since $b$ is a digit, $b$ can be 1. 6. **Check possible values:** - If $b=1$, then $a=2 \times 1 = 2$. - Check if digits are valid: $a=2$ (valid in base 4), $b=1$ (valid in base 7). 7. **Verify equality:** - $(ab)_4 = 4a + b = 4 \times 2 + 1 = 9$ - $(ba)_7 = 7b + a = 7 \times 1 + 2 = 9$ - They are equal. 8. **Find largest possible $a+b$:** - $a+b = 2 + 1 = 3$ **Final answer:** The largest possible value of $(a+b)_{10}$ is $3$ with $a=2$ and $b=1$.