8. Vusi makes bead bracelets in three different designs. The first design uses 23 beads, the second design uses 29 beads and the third design uses 31 beads. He has 750 beads and he wants to make the same number of bracelets using each design. How many beads will be left over?

To find the total number of beads Vusi needs for each design, you first need to determine the least common multiple (LCM) of the beads used in each bracelet design: 23, 29, and 31. Since these numbers are all prime, the LCM is their product: 23 × 29 × 31 = 21,169. However, Vusi only has 750 beads to work with. Now, let's determine how many complete sets of each bracelet design he can make with 750 beads. Since he needs a total of 23 + 29 + 31 = 83 beads per set, we can divide the total beads by the number of beads per complete set: 750 ÷ 83 ≈ 9 complete sets. This means Vusi can make 9 bracelets of each design. So, the total beads used will be 9 × 83 = 747 beads. Finally, subtract the used beads from the total beads he has: 750 - 747 = 3 beads left over. So, Vusi will have 3 beads left after making his bracelets.