Given:
\( a_n = 13 + 6(n - 1) \)
\( \Rightarrow a_n = 7 + 6n \)
Similarly, \( b_n = 15 + 7(n - 1) \)
\( \Rightarrow b_n = 8 + 7n \)
The common differences are 6 and 7, respectively.
The LCM of 6 and 7 is \( \text{LCM}(6, 7) = 42 \).
The first common term, determined by inspection, is 43.
A new arithmetic progression is formed starting at 43 with a common difference of 42:
\( t_m = 43 + (m - 1) \cdot 42 \).
We seek the largest term less than 1000:
\( 43 + (m - 1) \cdot 42 < 1000 \)
\( \Rightarrow (m - 1) \cdot 42 < 957 \)
\( \Rightarrow m - 1 < \frac{957}{42} \approx 22.78 \)
\( \Rightarrow m = 23 \).
Consequently, the 23rd term is:
\( t_{23} = 43 + (23 - 1) \cdot 42 = 43 + 22 \cdot 42 = 967 \).
Correct option: (C) 967
For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is: