13. a) It is known that among 80 coins there is one counterfeit coin, which is lighter than the others, which all have the same weight. Using four weighings on a balance scale without weights, find the counterfeit coin.

b) It is known that among $n$ coins there is one counterfeit coin, which is lighter than the others, which all have the same weight. What is the smallest number $k$ such that with $k$ weighings on a balance scale without weights, the counterfeit coin can always be identified?
13. a) Let's divide our coins into three groups: two groups of 27 coins each and one of 26 coins. During the first weighing, we will place the groups of 27 coins on the scales. If the scales do not balance, the counterfeit coin is on the "lighter" side. If the scales balance, the counterfeit coin is in the group of 26 coins. Thus, it is sufficient to learn how to solve the problem: to identify the counterfeit coin from a group of 27 coins using three weighings (the problem of identifying the counterfeit coin from a group of 26 coins can be reduced to this problem, for example, by adding one more arbitrary coin from the remaining 54 to the group of 26).

For the second weighing, we will divide the group of 27 coins into three groups of 9 coins each. By placing groups of 9 coins on both sides of the scales, we will find the group of 9 coins that contains the counterfeit coin.

By dividing the group of 9 coins, one of which is counterfeit, into three groups of 3 coins each, we will identify the group of 3 coins containing the counterfeit coin with the third weighing.

Finally, using the same method, we will find the counterfeit coin with the fourth weighing.

b) Let \( k \) be a natural number satisfying the inequalities \( 3^{k-1} < n \leq 3^k \). If \( n > 3^{k-1} \), then after \( k-1 \) weighings, the counterfeit coin will not be identified.

Remark. The answer to the problem can be briefly written in the following form: the minimum number of weighings required to identify the counterfeit coin from a group of \( n \) coins is \( \left[\log _{3}\left(n-\frac{1}{2}\right)+1\right] \), where the square brackets denote the integer part of the number (see p. 37).