7.4. On graph paper, a large square was drawn. It was cut into several identical medium squares. One of the medium squares was cut into several identical small squares. The sides of all squares run along the grid lines. Find the side lengths of the large, medium, and small squares if the sum of their areas is 154.
Answer: 12, 3, and 1 respectively.

Solution. From the problem statement, it follows that the side length of each square is a natural number, and the side length of each square is a divisor of the side length of the previous one. Let the side length of the small square be $a$, the medium square be $k a$, and the large square be $mka$. Then $(m k a)^{2}+(k a)^{2}+a^{2}=154 \Leftrightarrow$ $a^{2}\left(m^{2} k^{2}+k^{2}+1\right)=154$.

From the obtained equation, it follows that 154 is divisible by $a^{2}$. Since $154=2 \cdot 7 \cdot 11$, it is divisible only by $1^{2}$, so $a=1$. Then $k^{2}\left(m^{2}+1\right)=153$. Therefore, 153 is divisible by $k^{2}$. Considering that $153=3^{2} \cdot 17$ and $k>1$, we get: $k=3$. Substituting the found value of $k$ into the previous equation, we get that $m=4$. Thus, the side lengths of the squares are: small - 1, medium - 3, large - 12.

We can also form the equation $a^{2}+b^{2}+c^{2}=154$, where $a, b$, and $c$ are the sought side lengths, find all its natural solutions, and select the one that satisfies the condition. In this case, the enumeration must be complete and justified, in particular, the triplet $(9 ; 8 ; 3)$ should be found and discarded.

Grading criteria.

"+" A complete and justified solution is provided and the correct answer is obtained

“士" A generally correct reasoning is provided, containing minor gaps or inaccuracies, and the correct answer is obtained

“Ғ” The first equation is derived from divisibility considerations, but further conclusions are incorrect or absent

"Ғ" The correct answer is provided and it is verified that it satisfies the condition, or the correct answer is obtained by an incomplete enumeration

“-" Only the answer is provided

"-" An incorrect solution is provided or it is absent