6. The lengths of the sides of a rectangle are expressed as natural numbers (in cm), and the area is $196 \mathrm{~cm}^{2}$. How many such rectangles are there, and which one has the largest perimeter?
6. If we factorize the number 196 into prime factors, we have that $196=2 \cdot 2 \cdot 7 \cdot 7$.

3 points

From the prime factorization, it is clear that there are five different pairs of natural numbers whose product is 196, so we have five different rectangles with an area of $196 \mathrm{~cm}^{2}$, and these are: a rectangle whose adjacent side lengths are $1 \mathrm{~cm}$ and $196 \mathrm{~cm}, \quad 1$ point a rectangle whose adjacent side lengths are $2 \mathrm{~cm}$ and $98 \mathrm{~cm}$, 1 point a rectangle whose adjacent side lengths are $4 \mathrm{~cm}$ and $49 \mathrm{~cm}$, 1 point a rectangle whose adjacent side lengths are $7 \mathrm{~cm}$ and $28 \mathrm{~cm}$, 1 point a rectangle whose adjacent side lengths are $14 \mathrm{~cm}$ and $14 \mathrm{~cm}$. 1 point

The rectangle with the largest perimeter has adjacent side lengths of $1 \mathrm{~cm}$ and $196 \mathrm{~cm} .2$ points TOTAL 10 POINTS