Problem number 9. Four different positive integers are placed on a grid and then covered up. The products of the integers in each row and each column are shown in the diagram. What is the sum of the four integers? So first we can notice that in the first column then two numbers multiply to four and we can only write four as a product of two different numbers in two ways. One times four or four times one. Note that we cannot do two times two because all the numbers in the grid have to be different and so there are two possibilities. Either four is four times one, i.e. four is in the top left corner and one is in the bottom left corner or one is in the top left and four is in the bottom left. Let's try that first option. Here I've put four in the top left corner but what can fill in the question mark? We need four times something to be equal to six but there is no integer such that four times this integer equals six so this cannot be the correct option. The other possibility is that one is in the top left corner. In that case we must have four in the bottom left and six in the top right and now there is only one possibility for the number in the bottom right that will fill both four times something equals eight and six times something equals 12. That number is 2 and so the grid has been filled out completely. We want 1 plus 6 plus 4 plus 2 which is equal to 13 and so that means the answer is C, 13.

integer placement

grid puzzle

product conditions

sum calculation

positive integers