Numerical answer
How many terms are common to the arithmetic progressions 3, 7, 11, …, 407 and 2, 9, 16, …, 709?
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Answer: 14
Terms of the first AP are ≡ 3 (mod 4); terms of the second are ≡ 2 (mod 7).
The smallest number in both is 23 (23 = 3 + 4×5 = 2 + 7×3). Common terms then form an AP with difference LCM(4, 7) = 28: 23, 51, 79, …
Common terms cannot exceed 407 (the smaller last term): 23 + 28(k − 1) ≤ 407 → k − 1 ≤ 13.7 → k ≤ 14.
Number of common terms = 14 (the last is 23 + 28×13 = 387, which is also 2 + 7×55 ≤ 709).
Tip: find the first common term by listing a few terms of the AP with the larger difference.
The smallest number in both is 23 (23 = 3 + 4×5 = 2 + 7×3). Common terms then form an AP with difference LCM(4, 7) = 28: 23, 51, 79, …
Common terms cannot exceed 407 (the smaller last term): 23 + 28(k − 1) ≤ 407 → k − 1 ≤ 13.7 → k ≤ 14.
Number of common terms = 14 (the last is 23 + 28×13 = 387, which is also 2 + 7×55 ≤ 709).
Tip: find the first common term by listing a few terms of the AP with the larger difference.
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