From a series of numbers $1, 4, 7, 10, 13, \cdots.$, select seven that are consecutively arranged, their sum is a multiple of 11, the smallest of the seven selected numbers is at least
【Analysis and Solution】Arithmetic sequence.

Seven consecutive numbers form an arithmetic sequence with a common difference of 3; the sum of the seven numbers = the middle term $\times 7$;
Since the sum of these seven numbers is a multiple of 11;
Therefore, the middle term is a multiple of 11;
And these numbers are all 3 times a number +1;
The smallest number that satisfies both being a "multiple of 11" and "3 times a number +1" is 22;
That is, the middle term is at least 22;
The smallest number among the seven selected numbers is at least $22-(4-1) \times 3=13$.