Question

The number of common terms in the two sequences \(17, 21, 25, \ldots, 417\) and \(16, 21, 26, \ldots, 466\) is:

• A. 78
• B. 19
• C. 20
• D. 77
• E. 22

Hint:

For common terms of two APs, the common difference is the LCM of their differences, and the first common term can be found by checking small terms.

Answer 1

The Correct Option is C

First sequence: \(17, 21, 25, \ldots, 417\) Common difference \(d_1 = 4\), first term \(a_1 = 17\). Second sequence: \(16, 21, 26, \ldots, 466\) Common difference \(d_2 = 5\), first term \(a_2 = 16\). Common terms will form an arithmetic progression whose first term is the first common term: Checking, \(21\) is in both sequences. The common difference of the common terms is \(\text{LCM}(4,5) = 20\). Largest common term \(\leq 417\) and \(\leq 466\) is \(401\). Number of terms: \[ n = \frac{\text{last} - \text{first}}{\text{difference}} + 1 = \frac{401 - 21}{20} + 1 = \frac{380}{20} + 1 = 19 + 1 = 20 \] Thus, there are \(\boxed{20}\) common terms.