5. Five whole numbers are written around a circle. Two numbers that are next to each other, when added together, are never a multiple of three (a multiple of three is a number divisible by 3). Also, three numbers next to each other, when added together, are never a multiple of three.

How many of the five numbers are themselves a multiple of three?
A) 1
B) 2
C) 3
D) 4
E) Both 2 and 3 are possible
A5. B) 2 If three or more of the numbers are multiples of three, then there are two multiples of three next to each other, and the sum of these two numbers is a multiple of three. This is not allowed, so we conclude that there are at most two multiples of three in the circle.

Suppose there is one or no multiple of three in the circle. Then there are four consecutive numbers that are not multiples of three. These numbers cannot all give the same remainder when divided by 3, because otherwise the sum of three consecutive numbers would be divisible by 3. For example, if the numbers are 1, 4, 7, and 10, then $1+4+7=12$ is divisible by 3. Among the four numbers, there must be at least one that gives a remainder of 1 when divided by 3 and at least one that gives a remainder of 2 when divided by 3. We can specifically find two adjacent numbers where one gives a remainder of 1 and the other gives a remainder of 2 when divided by 3. Their sum is divisible by 3, and this is not allowed. We conclude that it is not possible for there to be one (or zero) multiple(s) of three in the circle.

It is indeed possible for there to be two multiples of three in the circle. For example, place 3, 4, 7, 6, and 1 in that order in the circle, and all conditions are met: $3+4, 4+7, 7+6, 6+1$, and $1+3$ are not divisible by 3, and neither are $3+4+7, 4+7+6, 7+6+1, 6+1+3$, and $1+3+4$.