3. Given 5 different natural numbers. The product of the two smallest of them is greater than 25, and the product of the two largest is less than 75. Find all these numbers (list all possible options and prove that there are no other options).
Solution. Let these numbers in ascending order be $a, b, c, d, e$. If $b$ is not greater than 5, then $a$ is not greater than 4, so $a b$ is not greater than 20. Therefore, $b$ is not less than 6. Similarly, if $d$ is not less than 9, then $e$ is not less than 10, and $d e$ is not less than 90. Therefore, $d$ is not greater than 8. But this is only possible when $b=6, c=7, d=8$. Now from the condition $25 / 6 < a < 6$ we get $a=5$, and from the condition $8 < e < 75 / 8$ we find $e=9$.