289. Find all integers $a$ for which the fraction

$$
\frac{a^{2}-21 a+17}{a}
$$

is an integer.
$\triangle$ Dividing the numerator of the fraction term by term by the denominator:

$$
\frac{a^{2}-21 a+17}{a}=a-21+\frac{17}{a}
$$

The representation of an algebraic fraction as the sum of a polynomial and a fraction with the same denominator, where the degree of the numerator is less than the degree of the denominator, is called the extraction of the integer part of the fraction; the polynomial is called the integer part of the fraction. In this case, the integer part of the fraction is $a-21$.

Since the difference $a-21$ takes integer values for all integer $a$, the question of the problem reduces to the following: for which integer values of $a$ is the fraction $\frac{17}{a}$ an integer?

To fully answer this question, we need to list all divisors of the number 17, including negative integers.

We get the numbers: $1, -1, 17, -17$.

Answer: $\pm 1, \pm 17$.