12. The numbers 1 to 8 are to be placed, one per circle, in the circles shown. The number next to each arrow shows what the product of the numbers in the circles on that straight line should be.
What will be the sum of the numbers in the three circles at the bottom of the diagram?
A 11
B 12
C 15
D 16
E 17
SOlUTION
$\mathbf{E}$

Let the numbers in each of the circles be $p, q, r, s, t, u, v$ and $w$, as shown in the diagram. Since the only two lines of numbers with products divisible by 5 meet at the circle containing letter $r$, we have $r=5$. Similarly, since the only two lines of numbers with products divisible by 7 meet at the circle containing letter $v$, we have $v=7$. Now consider the line of numbers with product 28. Since we know $v=7$, we have $p \times s=4$ and, since the numbers are all different, $p$ and $s$ are some combination of 1 and 4.
Now note that 4 is not a factor of 30 and so $p$ cannot be 4 and hence $p=1$ and $s=4$. It is now easy to see that the only way to complete the diagram is to put $q=6, t=3, u=2$ and $w=8$. Therefore the sum of the numbers in the bottom three circles is $2+7+8=17$.