Jenny is making bracelets using different colored

beads. Jenny has 60 green beads and 105 blue

beads. What is the greatest number of identical

bracelets Jenny can make if she wants to use all of

the beads?

Answer:Jenny 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.

Answer