How many terms are common to the two arithmetic progressions 3, 7, 11, …, 407 and 2, 9, 16, …, 709?

Terms of the first AP are ≡ 3 (mod 4); terms of the second are ≡ 2 (mod 7).

The smallest number satisfying both: check 2, 9, 16, 23 … → 23 (23 = 4×5 + 3 ✓). Common terms form an AP with first term 23 and common difference LCM(4, 7) = 28.

They must not exceed the smaller last term, 407: 23 + 28(k − 1) ≤ 407 → 28(k − 1) ≤ 384 → k − 1 ≤ 13.7 → k = 14.

The common terms are 23, 51, …, 387: 14 terms. (The next, 415, exceeds 407.)