A necklace is formed by stringing 133 colored beads on a thin wire in the following order: red, orange, yellow, green, blue, indigo, violet; red, orange, green, blue, indigo, violet. If the pattern continues, what will the color of the 101st bead on the string?

The seven colors keep repeating in the same order.

Color: red orange yellow green blue indigo violet

Bead

Number: 1 2 3 4 5 6 7

8 9 10 11 12 13 14 15 etc.

The violet beads are in positions 7,14,21 ...,70, ..., that is multiples of 7.

If 101 beads were a multiple of 7, the 101st bead would be violet, but when 101 is divided by 7, the quotient would be 14 and a remainder of 3.

Since 14*7=98, the 98th bead completes the 14th cycle, and hence is violet.

The 99 starts the next cycle; is is red.

The 100th bead is Orange, and the 101st bead is yellow.

Answer: Yellow