Bryce has two pieces of yarn, one 45 feet long and the other 15 feet long. For a craft project, he wants to cut them up to produce many pieces of yarn that are all of the same length, with no yarn left over. What is the greatest length, in feet, that he can make them?

Answer 1

To find the greatest length of yarn pieces that Bryce can produce from the two pieces of yarn (one 45 feet long and one 15 feet long) without leaving any leftover yarn, we need to determine the greatest common divisor (GCD) of the two lengths.

We start by listing the positive divisors of each length.

Divisors of 45:

1. 1
2. 3
3. 5
4. 9
5. 15
6. 45

Divisors of 15:

1. 1
2. 3
3. 5
4. 15

Now we identify the common divisors between the lists of divisors of 45 and 15:

• Common divisors: 1, 3, 5, 15

The greatest common divisor (GCD) of 45 and 15, which is the largest of the common divisors, is 15.

To confirm, we can also find the GCD using the Euclidean algorithm:

1. Begin with the lengths: 45 (larger) and 15 (smaller).
2. Divide 45 by 15, which gives a quotient of 3 and a remainder of 0.
3. Since the remainder is 0, we conclude that the GCD is the last non-zero remainder, which is 15.

Thus, the greatest length, in feet, that Bryce can cut all pieces of yarn into, without leaving any leftover, is

\[ \boxed{15} \]