Matthew has four pieces of wood with length 160 centimeters, 0.8 meters, 40 centimeters, and 2.4 meters. He wants to cut all the lengths of wood into equal-sized pieces without having any left over. By finding the GCF, determine the greatest possible length of each piece.
GCF stands for greatest common factor. The greatest common factor will give the greatest possible length of each piece that Matthew should use as this number will be the biggest number that divides into all four lengths. This will mean that the complete length of each piece of wood is used so there won’t be any wastage.
The full length has been expressed using a mix of units. Two of them were expressed in centimeters and two were expressed in meters. We first need to convert to a common unit. To do so, we need to recall the conversion between meters and centimeters, which is that one meter is equivalent to 100 centimeters. I’m going to convert the two lengths that have been expressed in meters into centimeters. 0.8 meters first of all is equal to 0.8 multiplied by 100 centimeters. It’s equivalent to 80 centimeters. 2.4 meters can be converted in exactly the same way. I multiply it by 100, giving 240 centimeters.
Next, I need to consider how to find the greatest common factor of these four numbers. And to do so, I’m going to use factor trees. Let’s start with the number 40. I need to consider how I can break this down into a product. And I’ll begin with the product of four and 10. This gives me my first branch of the factor tree. 40 can be written as four multiplied by 10. Neither of these are prime numbers. So I need to continue with the factor tree. Four can be written as the product of two and two and 10 can be written as the product of five and two. Two and five are both prime numbers, which means the factor tree can’t continue any further.
I found all the prime factors of 40. Next, I need to repeat this for the other three lengths. 160 first of all can be written as the product of 16 and 10. Neither of these are prime numbers. So I need to continue the factor tree. 16 can be written as the product of eight and two and 10 can be written as the product of five and two. Two and five are prime numbers. So these branches of the factor tree stop here. However, eight isn’t. So I need to continue this branch. Eight can be written as the product of four and two, with two being a prime number. Four, however, needs to be broken down further into the product of two and two.
I’ve now found all the prime factors of 160. You can now pause the video and draw factor trees for the remaining two lengths: 80 centimeters and 240 centimeters. Here are my factor trees for 240 and 80. Yours might look slightly different and that’s okay. For example, you might have decided to break 240 down to the product of four and 60 first of all. This will mean that the branches of your factor tree looks slightly different. But the prime numbers that you reach at the end of the branches should be the same or a bit in a different order.
Now that we have the four factor trees, we need to remember how to use these factor trees to find the greatest common factor of the four numbers. The greatest common factor will be equal to the product of all of the factors that the four numbers have in common. So we need to identify common factors.
All four factor trees have a common factor of two first of all. They also share a second factor of two. In fact, they also share a third factor of two. 160, 240, and 80 also share a fourth factor of two. But as this isn’t shared with 40, it isn’t a common factor of all four numbers. Looking at the factor trees again, we can see that there is also a common factor of five shared between all four numbers. Are there any other common factors? Well, they can’t be because we’ve actually used all of the factors of 40.
The greatest common factor of the four numbers then is therefore found by multiplying the four common factors together. Two multiplied by two multiplied by two multiplied by five. The product of these four numbers is 40. You may have been able to spot that already as 40, 80, 160, and 240 are all fairly familiar multiples of 40.
The greatest possible length of each piece that Matthew should use when cutting this wood is 40 centimeters.