10.2. Can the signs $*$ in the expression $\frac{1}{2} * \frac{1}{3} * \frac{1}{4} * \frac{1}{5} * \frac{1}{6} * \frac{1}{7} * \frac{1}{8}$ be replaced with plus and minus signs so that the absolute value of this expression becomes less than $\frac{1}{500}$?
Solution: The least common multiple of all the denominator numbers is 840. If the numerator after converting all fractions to this common denominator is $0, -1$, or $1$, the answer is positive; otherwise, it is negative. The numerators of the fractions after conversion are $420, 280, 210, 168, 140, 120$, and 105. The last digit of their sum (considering the signs of the numbers) will be 3 or 7. Therefore, it is impossible to arrange the signs in this way.

Answer: It is impossible.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct justified answer | 7 points |
| Proven that after converting to a common denominator, the numerator of the resulting fraction will be odd (or not equal to $0$) | 3 points |
| Proven that after converting to a common denominator, the problem requires obtaining an irreducible fraction with a numerator not exceeding the absolute value of 1 | 1 point |
| Incorrect answer or correct answer without justification, possibly illustrated with several examples | 0 points |