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Let's denote the smallest odd integer as x. The next three consecutive odd integers would then be x+2, x+4, and x+6. The sum of these four integers is x + (x+2) + (x+4) + (x+6) = 4x + 12. Given that the sum is 112, we have 4x + 12 = 112. Solving for x, we get x = 25. Therefore, the greatest of the four integers is x + 6 = 25 + 6 = 31.

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