We are given that the sum of the two consecutive odd integers is at most 100, so we can write:
x + (x+2) ≤ 100
Simplifying the above inequality, we get:
2x + 2 ≤ 100
2x ≤ 98
x ≤ 49
Since x is an odd integer, the largest possible value of x is 49. Therefore, the two consecutive odd integers are 49 and 51, and their sum is 100, which is the maximum possible sum of two consecutive odd integers that satisfy the given condition.
Hence, the pair of consecutive odd integers with the greatest sum is (49, 51).
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