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?
90gb

Cameron can make  

18

identical necklaces, each containing 5 green and 6

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

a) 7x=7

step-by-step explanation: