5. A right-angled triangle is called Pythagorean if the lengths of all its sides are natural numbers. Find the greatest integer that divides the product of the lengths of the sides of any Pythagorean triangle.
# Problem 5.

Answer: 60.

Solution. Since a triangle with sides $3,4,5$ is a right triangle, the desired number cannot be greater than 60. We will show that the product of the lengths of the sides of any Pythagorean triangle is divisible by 60, i.e., by 3, by 4, and by 5.

The remainders when the square of an integer is divided by 3 and 4 can only be 0 and 1. Therefore, if $a$ and $b$ are not divisible by 3 and 4, then $a^{2}+b^{2}$, when divided by 3 and 4, gives a remainder of 2, and thus cannot be a perfect square.

The remainders when an integer is divided by 5 can only be 0, 1, and 4. Therefore, if $a$ and $b$ are not divisible by 5 and $a^{2}+b^{2}$ is a perfect square, then one of the numbers - $a^{2}$ or $b^{2}$ - has a remainder of 1, and the other has a remainder of 4. Consequently, $a^{2}+b^{2}$ is divisible by 5.