For which values of $a$ will the expression

$$
97 a^{2}+84 a-55
$$

be a multiple of $a$?
We say that an integer $c$ is a multiple of an integer $d$ (or that $c$ is divisible by $d$, or that $d$ is a divisor of $c$), if there is an integer $q$ such that $c = d q$. In questions of divisibility, it may be useful to focus on positive divisors, that is, to assume that $d$ is a natural number, or perhaps that every number in the definition is a natural number, while at other times it may be useful to allow all integers.

Thus, the question to be examined is which integers, or positive integers, have an integer $k$ for which the following equality holds:

$$
97 a^{2} + 84 a - 55 = k a
$$

Rearranging the terms containing $a$ to one side and factoring out $a$,

$$
(97 a + 84 - k) a = 55
$$

Accordingly, 55 must be expressed as the product of two positive integers, or two integers. The positive integer factorizations are $1 \cdot 55, 5 \cdot 11$, and the factorizations resulting from swapping the factors, while the other integer factorizations arise by taking the negatives of both factors of the positive factorizations. Thus, the following positive integer values of $a$ are solutions: $1, 5, 11, 55$, and considering all integer values, their negatives are added: $-1, -5, -11, -55$.

Remark: Béla Bollobás also raised the more general question of what values of $a$, without assuming that $a$ is an integer, make the given expression an integer multiple of $a$, that is, for what (not necessarily integer) values of $a$ does the following hold with an integer $k$. The wording of the problem allows for this interpretation, but this question cannot be addressed in the same way. The question now is for what values of $a$ does the following hold with an integer $k$:

$$
97 a^{2} + (84 - k) a - 55 = 0
$$

Here, $k$ is an integer if and only if the number $84 - k = l$ is an integer. Introducing this notation, the numbers

$$
a = \frac{-l \pm \sqrt{l^{2} + 21340}}{194}, \quad (l = 0, \pm 1, \pm 2, \ldots)
$$

and only these numbers satisfy the requirements set forth.