Problem 10.7. Oleg has four cards, on each of which natural numbers are written on both sides (a total of 8 numbers are written). He considers all possible quadruples of numbers, where the first number is written on the first card, the second on the second, the third on the third, and the fourth on the fourth. Then, for each quadruple, he writes down the product of the numbers in his notebook. What is the sum of the eight numbers on the cards if the sum of the sixteen numbers in Oleg's notebook is $330 ?$
Answer: 21.

Solution. Let the numbers on one card be $a$ and $b$, on another card - $c$ and $d$, on the third card - $e$ and $f$, and on the fourth card - $g$ and $h$. According to the problem, the sum of 16 terms of the form $a c e g + a c e h + \ldots + b d f h$ equals 330. Note that this sum is also obtained by expanding all the brackets in the expression $(a+b)(c+d)(e+f)(g+h)$.

Therefore, $(a+b)(c+d)(e+f)(g+h)=330=2 \cdot 3 \cdot 5 \cdot 11$. Since all numbers are natural, each bracket is greater than 1. Hence, the brackets are equal to the numbers $2, 3, 5$, and 11 in some order. Then their sum is $a+b+c+d+e+f+g+h=2+3+5+11=21$.