How many terms are identical in the two APs 19, 21, 23, \dots up to 110 terms and 19, 22, 25, 28, \dots up to 75 terms?

Explanation

First AP: d_1 = 2, max term = 19 + 109 \times 2 = 237. Second AP: d_2 = 3, max term = 19 + 74 \times 3 = 241. The identical terms form a new AP with a = 19 and common difference d = \text{lcm}(2, 3) = 6. We need 19 + 6(n - 1) \leq 237 \implies 6(n - 1) \leq 218 \implies n - 1 \leq 36.33. Therefore, n = 37.

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