A necklace is made by stringing N individual beads together

Here's how the necklace looks when it is strung:

RED-GREEN-WHITE-BLUE-YELLOW

We know that the length of the pattern is 5 beads (shown above). The necklace begins at the red bead and then ends in a white bead. The shortest necklace we could have would be RED-GREEN-WHITE, which would only be 3 beads and then fit the bill. The next shortest necklace we could have would be when the pattern of 5 beads has been repeated again. In other words (3+5=8 beads) In this case the pattern would loop once (5 beads), then start RED(6)-GREEN(7)-WHITE(8).

The pattern that emerges here is that the number of beads will be in this list:

3
3+5 = 8
3+5+5 = 13
3+5+5+5 = 18
etc

Any easier way of writing this pattern would be to say that the number of beads must be divisible by 5 with 3 left over. or, when the number is divided by 5 it has a remainder of 3. Then we move onto our choices:

16/5 = 3 remainder 1
32/5 = 6 remainder 2
41/5 = 8 remainder 1
54/5 = 10 remainder 4
68/5 = 13 remainder 3

BINGO! We see that 68 is our answer.

an even shorter method would be to realize that the number must end in 3 or 8 because multiples of 5 end in 0 or 5. Thus, the only choice that ends in 3 or 8 would be our answer!

Answer E