Starting with positive integer M written on the board, Alice plays the following game: in each move, if is the number on the board, she replaces it with 3x + 2. Similarly, starting with positive integer N written on the board, Bob plays the following game: in each move, if is the number on the board, he replaces it with 2x + 27. Given that Alice and Bob reach the same number after playing 4 moves each, find the smallest value of M + N.
Hint 1: Start by analyzing the initial conditions and setting up the basic equations. In 4 steps Alice will write.
Hint 2: Look for algebraic properties, symmetry, or geometric theorems to simplify. (3(3(3(3M + 2) + 2) + 2) + 2) = .
Hint 3: Proceed with the final algebraic steps to solve the system. and Bob will write.
Step 1: In 4 steps Alice will write
Step 2: (3(3(3(3M + 2) + 2) + 2) + 2) =
Step 3: and Bob will write
Step 4: (2 (2 (2 (2N + 27) + 27) + 27) + 27) =
Step 5: As = , we get
Step 6: 81M = 16N + 325
Step 7: => Mmin = 5 and Nmin = 5
Step 8: (M + N)min = 10
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