6. The numbers $5,6,7,8,9,10$ are to be placed, one in each of the circles in the diagram, so that the sum of the numbers in each pair of touching circles is a prime number. The number 5 is placed in the top circle.
Which number is placed in the shaded circle?
A 6
B 7
C 8
D 9
E 10
Solution
E

The primes that are the sum of two numbers chosen from $5,6,7,8,9$ and 10 are all greater than 2, and therefore must be odd primes. So if they are the sum of two integers, one of these must be odd and the other even.
It follows that of the two numbers in any pair of touching circles, one must be odd and one must be even. So the numbers must be arranged in the circles so that they are alternately odd and even. Therefore the number in the shaded circle must be even, that is, 6,8 or 10 .

We also deduce from this that the only possible positions for the even numbers not in the shaded circle are the circles adjacent to the top circle which contains the number 5 .
Now the number 10 cannot be adjacent to the top circle because $5+10=15$ which is not a prime. Therefore 10 must be the number in the bottom circle.
Commentary
In the context of the SMC it is adequate to stop here. For a full solution we need to show that it is possible to put the numbers $6,7,8,9$ and 10 in the circles in a way that meets the requirement that the sum of the numbers in each pair of touching circles is prime.

To complete the solution we show that there actually is an arrangement with the number 10 in the bottom circle. It is easy to see that the two arrangements shown in the figures below both meet the requirement that the sum of the numbers in each pair of touching circles is a prime. Note that the only difference between them is that in the figure on the left the numbers 5, 6, 7, $10,9,8$ go round clockwise, whereas in the figure on the right they go round anticlockwise.
For investigation
6.1 Show that the two arrangements of the numbers given in the above solution are the only arrangements that meet the requirements of the question.