7.5. There are 200 matches. How many ways are there to form, using all the matches, a square and (separately) an equilateral triangle? (Different ways differ in the sizes of the square and the triangle).
Answer: 16. Solution. Let $x$ (matches) be the side length of the square, and $y$ (matches) be the side length of the triangle. Then $4 x+3 y=200 \Leftrightarrow 3 y=4(50-x)$. Thus, for divisibility by 3, we need to consider all natural numbers up to 50 as $x$ that give a remainder of 2 when divided by 3 (i.e., the same remainder as the number 50). We have 16 such numbers, namely numbers of the form $2+3 k, k=0,1, \ldots, 15$, from 2 to 47 (the number 50 itself should not be considered, as otherwise only a square would be formed, without a triangle). Then the corresponding value of $y$ is $4 \cdot(16-k)$.