Let m, be natural numbers such that + 3n – 5 = 5LCM(m, n) – 11GCD(m, n). Find the maximum possible value of + n.
Hint 1: Start by analyzing the initial conditions and setting up the basic equations. Let G. C. D. of (m, n) = d.
Hint 2: Look for algebraic properties, symmetry, or geometric theorems to simplify. Then for some positive coprime integers and y.
Hint 3: Proceed with the final algebraic steps to solve the system. m = dx and = dy.
Step 1: Let G. C. D. of (m, n) = d
Step 2: Then for some positive coprime integers and y
Step 3: m = dx and = dy
Step 4: + 3n – 5 = 2 LCM (m, n) – 11 GCD (m, n)
Step 5: => dx + 3dy – 5 = 2dxy – 11d
Step 6: or, d(x + 3y – 2xy + 11) = 5
Step 7: Now, to maximize the sum + n, must be 5
Step 8: => + 3y – 2xy + 11 = 1
Step 9: or + 3y – 2xy + 10 = 0
Step 10: or (2x – 3)(2y – 1) = 23
Step 11: Case I : 2x – 3 = 1 and 2y – 1 = 23
Step 12: => = 2 and = 12
Step 13: This is not possible as x, are coprime
Step 14: Case II : 2x – 3 = 23 and 2y – 1 = 1
Step 15: => = 13 and = 1
Step 16: => + = (13 + 1) × 5 = 70
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