1. **Problem statement:** We are given 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 $(a+b)_{10}$ such that $(ab)_4 = (ba)_7$, and find non-zero values of $a$ and $b$ that satisfy this equality or prove none exist. 2. **Express the numbers in base 10:** - $(ab)_4 = 4a + b$ - $(ba)_7 = 7b + a$ 3. **Set the equality:** $$4a + b = 7b + a$$ 4. **Simplify the equation:** $$4a + b = 7b + a \\ 4a - a = 7b - b \\ 3a = 6b \\ a = 2b$$ 5. **Constraints on digits:** - Since $(ab)_4$ is base 4, digits $a,b$ must be in $\{0,1,2,3\}$. - Since $(ba)_7$ is base 7, digits $a,b$ must be in $\{0,1,2,3,4,5,6\}$. - Both $a$ and $b$ are non-zero. 6. **Find valid pairs $(a,b)$:** - From $a=2b$, and $a,b \in \{1,2,3\}$ (since $a$ must be at most 3 for base 4), possible $b$ values are 1 or 2: - If $b=1$, then $a=2$ (valid) - If $b=2$, then $a=4$ (invalid for base 4) 7. **Check the pair $(a,b) = (2,1)$:** - $(ab)_4 = 4 \times 2 + 1 = 8 + 1 = 9$ - $(ba)_7 = 7 \times 1 + 2 = 7 + 2 = 9$ - Equality holds. 8. **Calculate $(a+b)_{10}$:** $$a + b = 2 + 1 = 3$$ 9. **Conclusion:** - The largest possible value of $(a+b)_{10}$ is 3. - The non-zero digits satisfying $(ab)_4 = (ba)_7$ are $a=2$ and $b=1$. **Final answer:** $$\boxed{3}$$