1. Upon entering the Earth's atmosphere, the asteroid heated up significantly and exploded near the surface, breaking into a large number of fragments. Scientists collected all the fragments and divided them into groups based on size. It was found that one-fifth of all fragments had a diameter of 1 to 3 meters, another 26 were very large (more than 3 meters in diameter), and the rest were divided into several groups, each of which accounted for 1/7 of the total number of fragments. How many fragments did the scientists collect?
1. Answer: 70. Solution. Let $\mathrm{X}$ be the total number of fragments. The condition of the problem leads to the equation:

$\frac{x}{5}+26+n \cdot \frac{X}{7}=X$, where $n-$ is the unknown number of groups. From the condition of the problem, it follows that the number of fragments is a multiple of 35

$$
X=35 l, l \in \mathbb{N}
$$

Then the first equation can be rewritten in another form: $7 l+26+n \cdot 5 l=35 l$. From this, we express: $5 n=28-\frac{26}{l}$. It is clear that the natural variable $l$ can take the values $1,2,13$ and 26. By direct verification, we find that only one variant is possible $l=2, n=$ 3. Substituting into (1) we get the answer $X=70$.