Exercise 12 Page 38

We have 16 yellow beads, 20 red beads, and 24 orange beads that we want to use to make identical bracelets. We want to find the greatest number of bracelets we can make if we use all the beads. To do so, we will start by listing the factor pairs of each number of beads. Let's start with the factor pairs of 16. 16 &= 1 * 16 16 &= 2 * 8 16 &= 4 * 4 There are 5 possible arrangements for yellow beads.

• 1 bracelet with 16 yellow beads.
• 2 bracelets with 8 yellow beads.
• 4 bracelets with 4 yellow bead.
• 16 bracelets with 1 yellow beads.
• 8 bracelets with 2 yellow beads.

  Now let's list the factor pairs of 20. 20 &= 1 * 20 20 &= 2 * 10 20 &= 4 * 5 This means that there are 6 arrangements for red beads.

  • 1 bracelet with 20 red beads.
  • 2 bracelets with 10 red beads.
  • 4 bracelets with 5 red beads.
  • 20 bracelets with 1 red bead.
  • 10 bracelets with 2 red beads.
  • 5 bracelets with 4 red beads.

  Finally, let's list the factor pairs of 24. 24 &= 1 * 24 24 &= 2 * 12 24 &= 3 * 8 24 &= 4 * 6 This means that there are 8 arrangements for orange beads.

  • 1 bracelet with 24 orange beads.
  • 2 bracelets with 12 orange beads.
  • 3 bracelets with 8 orange beads.
  • 4 bracelets with 6 orange beads.
  • 24 bracelets with 1 orange bead.
  • 12 bracelets with 2 orange beads.
  • 8 bracelets with 3 orange beads.
  • 6 bracelets with 4 orange beads.

  The numbers that make up the factor pairs are the factors of each color of bead. Factors of 16&: 1,2, 4,8,16 Factors of 20&: 1,2, 4,5,10,20 Factors of 24&: 1,2,3, 4,6,8,12,24 As we can see, the greatest common factor between the three numbers is 4. This means that the greatest number of bracelets that we can make with no beads leftover is 4.

  • 4 bracelets with 6 orange beads.
  • 4 bracelets with 5 red beads.
  • 4 bracelets with 4 yellow bead.

  Therefore, we can make at most 4 bracelets using all the beads.