57. There are 90 equally spaced dots marked on a circle. Shannon chooses an integer, n. Beginning at a randomly chosen dot, Shannon goes around the circle clockwise and colors in every nth dot. He continues going around and around the circle coloring in every nth dot, counting each dot whether it is colored in or not, until he has colored in every dot. Which of the following could have been Shannon's integer n ?

Answer and Explanation

Your Answer is

Correct Answer is E

Explanation

90 is divisible by 3, 5, and 6, so it will cycle after the first round, and there must be points that are not colored.

90÷4 has a remainder, and it will cycle after the second lap. That is, 90×2 is divisible by 4. 90×2÷4=45, that is, half of the points are colored and half of the points are not colored.

90÷7 has a remainder, and after the second and third rounds, it still cannot be cycled, that is, neither 90×2 nor 90×3 can be divisible by 7. It cannot be divisible until after the seventh round, 90×7÷7=90. At this point all 90 points are colored.