Using the GCF (Greatest Common Factor) to Solve Problems
In today’s post we’re going to look at a series of problems that can be solved by using factorization and the GCF (Greatest Common Factor)
First we’re going to look back at previous blog entries to remind ourselves what the GCF is, and how we calculate the GCF of two numbers:
After you’ve refreshed your memory, we’re going to put what we know into practice with an example. After that, we’ll pose two problems for you to solve yourself. Don’t forget to leave us your answer in the comments once you’ve done them!
Example: Solving a problem using the GCF
Rosie and Mark are twins and today is their birthday. Rosie has brought 24 gummy bears to share in class and Mark has brought 18 candy bars. If they want to share their candy with their friends in such a way that all their friends have the same amount of each type of candy and they give out as much candy as possible, how many friends could they give candy to?
The problem requires us to find the maximum quantity of equal groups that we can make with the gummy bears and candy bars, making sure both twins give out the same number of groups. That way, each group will contain the same quantity of gummy bears and candy bars.
To solve this problem, the first thing we have to do is decompose both numbers into prime factors.
24 = 23 x 3
18 = 2 x 32
Now, to calculate the GCF, we need to choose the common factors with the smallest exponent, which in this case will be the 2 and the 3.
Once we’ve chosen our common factors, all we have to do is multiply them by each other. 2 multiplied by 3 equals 6 (2 x 3 = 6).
Therefore, the GCF of 24 and 18 is 6
This means they can make a maximum of 6 groups of each type of candy. 24 gummy bears in 6 groups makes 4 gummy bears in each group, and 18 candy bars in 6 groups makes 3 candy bars in each group.
So, each one of the 6 friends that are going to get candy from Rosie and Mark are going to get 4 gummy bears and 3 candy bars.
Now you know how it’s done, see if you can solve the following problems without any help!
Problem 1: Practice using the GCF
Andy has two ropes. One is 120 feet long, and the other one is 96 feet long. He wants to cut them in such a way that all the pieces are equal, but as long as possible. How many pieces of rope will he get?
Problem 2: Practice using the GCF
A shop buys USB flash drives of different colors wholesale. For Christmas, they made a special order of 84 red flash drives, 196 blue ones, and 252 green ones. To help them store the merchandise neatly, they asked that the flash drives be sent in equal boxes, without mixing any colors, and with each box containing the greatest number of flash drives possible.
If the order is sent in the way the shop requested, how many flash drives will there be in each box, and how many boxes of each color will there be?
I hope these problems have helped you practice and improve your ability to calculate using the GCF!
And don’t forget to leave the solution to each problem in the comments!
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