1. Problem 37: Given digits A and B, two-digit numbers AB and 5A satisfy the equation $$AB \times 3 = 5A$$. Find $$A^2 + B^2$$. 2. Let the two-digit number AB be $$10A + B$$ and 5A be $$50 + A$$. 3. The equation becomes $$3(10A + B) = 50 + A$$. 4. Expanding: $$30A + 3B = 50 + A$$. 5. Rearranging: $$30A - A + 3B = 50$$ which simplifies to $$29A + 3B = 50$$. 6. Since A and B are digits, $$A \in \{1,...,9\}$$ and $$B \in \{0,...,9\}$$. 7. Try values of A to find integer B: - For $$A=1$$: $$29(1) + 3B = 50 \Rightarrow 3B = 21 \Rightarrow B=7$$ (valid digit) 8. So, $$A=1$$ and $$B=7$$. 9. Calculate $$A^2 + B^2 = 1^2 + 7^2 = 1 + 49 = 50$$. Final answer: 50.