3. Represent the fraction $\frac{179}{140}$ as a sum of three positive fractions with single-digit denominators. Explain your answer!
Solution. If we factorize the denominator of the given fraction into prime factors, we get $140=2 \cdot 2 \cdot 5 \cdot 7$. From this, this number can be written as a product of three single-digit numbers in only the following way: $140=4 \cdot 5 \cdot 7$. Therefore, the required representation of the given fraction is $\frac{x}{4}+\frac{y}{5}+\frac{z}{7}=\frac{179}{140}$. If we multiply the equation by 140, we get $35 x+28 y+20 z=179$.

It is sufficient to find all solutions to this equation. First, note that the number $20z$ is an even number ending in 0 for any value of $z$. Then, the number $28y$ is even for any value of $y$. Since the right side of the equation is an odd number, the number $35x$ must be odd, which is possible only if $x$ is an odd number. Furthermore, $35x$ ends in 5 for any value of $x$, so $28y$ must end in 4. This is possible if $y$ is a number ending in 3 or 8. If $y$ is 8, then $28y>179$. Therefore, we conclude that $y=3$. We have

$$
35 x+20 z=95
$$

By direct verification, we get $x=1, z=3$, that is,

$$
\frac{1}{4}+\frac{3}{5}+\frac{3}{7}=\frac{179}{140}
$$