Question

Makayla has two types of ropes. She wants to cut the ropes into pieces of the same length for butterfly knots. One is 84 inches, the other is 116 inches.

Find the greatest possible length that she can cut for each piece, so that no rope will be left unused.

Answer 1

In this problem, we are trying to find the greatest common factor, or the greatest amount that both values share. To find this, we will need to factor our values though prime factorization:

`84 = 2^2 \cdot 7^1 \cdot 3^1`

`116 = 2^2 \cdot 29`

We can see that the greatest value that we can find in both numbers is
`2^2`, or 4, meaning our answer is 4 inches.

Answer 2

Remark

If you want every piece you cut equal to the same length, you are talking about the GCF because both ropes have to be cut in such a way that all parts of the whole set of ropes are the same size.

Prime Factors of 84 and 116

84: 2 * 2 * 3 * 7

116: 2 * 2 * 29

The two 2s are all that is common.

So that means that the 84 inch length will give you 21 lengths that are 4 inches long and the 116 inch length will give you 29 lengths that are 4 inches long.