Let a, be positive integers satisfying a3 – b3 – ab = 25. Find the largest possible value of a2 + b3.
Hint 1: Start by analyzing the initial conditions and setting up the basic equations. a3 – b3 – ab = 25 for = 4 and = 3.
Hint 2: Look for algebraic properties, symmetry, or geometric theorems to simplify. Because for any greater number a3 – b3 – ab > 25.
Hint 3: Proceed with the final algebraic steps to solve the system. To prove this if > b, then a3 – b3 – ab.
Step 1: a3 – b3 – ab = 25 for = 4 and = 3
Step 2: Because for any greater number a3 – b3 – ab > 25
Step 3: To prove this if > b, then a3 – b3 – ab
Step 4: = (b + t)3 – b3 – b(b + t), > 0
Step 5: = (3t – 1)b2 + (3t2 – t)b + t3 is always greater than 4,
Step 6: then 3
Step 7: So, a2 + b3 = 42 + 33 = 43
Ready to track your progress and master these topics?
Create a free account