Question

Cameron is making bead necklaces. He has 90 green beads and 108 blue beads. What is the greatest number of identical necklaces he can make if he wants to use all of the beads?

Answer 1

Cameron can make `18` identical necklaces, each containing 5 green and 6 blue beads.

Explanation:

Assume, each necklace contains `G` green and `B` blue beads and we have `N` such necklaces. All these variables are natural numbers.

Then we can establish the following equations in natural numbers:
`N*G = 90`
`N*B = 108`

Our task is to find a maximum `N` for which these two equations have a solution in natural numbers.
Obviously, `N` is a maximum common denominator of `90` and `108`.

To find the maximum common denominator of `90` and `108`, let's represent these two numbers as a product of prime numbers:
`90 = 2*3*3*5`
`108 = 2*2*3*3*3`
As we see, the maximum common denominator (a product of all prime numbers that are identical for both `90` and `108`) is
`P=2*3*3 = 18`

Therefore, assigning `N=18`, `G=5` and `B=6`, we obtain the solution:
Maximum number of identical necklaces is `N=18` with each necklace containing `G=5` green beads and `B=6` blue beads.