In this article, you will certainly learn exactly how to calculation the Greatest usual Factor (GCF) and Least typical Multiple (LCM) that integers. These an abilities are beneficial in a variety of various situations. Let’s view what they room all about!
Suppose her mom has actually two rectangle-shaped cakes for a party, one vanilla and also one strawberry, both the very same thickness. The vanilla flavored cake has an area that 12 square inches, when the strawberry flavored cake has actually an area the 20 square inches. Your mother wants to division both cakes into pieces such the all pieces space of equal size and both cakes space evenly split with no leftover smaller sized pieces.
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“No problem,” your mom says, “I’ll just cut the cakes right into 4-square-inch pieces. The way, anyone will have actually a same-sized piece.”
How in the world did she know that?
More GCF and also LCM Help
The Greatest typical Factor
Before we discover the cake situation, stop remember what factors are. Factors are number that, as soon as multiplied, produce an additional number. Usually, we talk about factors in pairs. 1 and also 8 are determinants of 8 because 1×8=8. The number 4 and 2 are likewise factors the 8, because 4×2=8.
Numbers have a unique prime factorization. Recall the prime numbers, favor 2, 3, and 5, have the right to only be split by themselves and also 1. We deserve to keep dividing the components of any kind of number into its element number components. So, any type of number is written of a unique set of element numbers multiplied together. Take a look at at this factor trees and what they phone call us:
Factorizations of 18
The number 18 is originally factored two different ways: 3 and 6 and likewise 2 and 9. We have the right to see the prime factorization is the same, regardless of our very first factor pair.
2 x 3 x 3 = 18. The purple circles match in both trees.
What go this need to do v the two cakes?
Mathematically, your mommy calculated the Greatest common Factor (GCF) in between 12 and also 20 to be 4. In other words, 4 is the largest factor that both numbers have actually in common. We can see the by looking in ~ the prime factorizations the both numbers:
The blue numbers room the prime factors of both numbers. To discover the GCF, simply determine the prime determinants that both numbers have actually in common and also multiply castle together.
Both number have usual prime factors of 2 and 2. 2 x 2 = 4. This describes how your mom knew to reduced both cakes into 4-square-inch pieces!
What is the GCF the 15 and also 27?
In this case, both number only have actually one typical factor, 3. The equipment is frequently written GCF (15,27) = 3.
What is the GCF that 18 and 36?
GCF (18,36) = 2 x 3 = 6.
Find the GCF that 7 and also 56.
In this case, among the number is prime and it is a aspect of the other number. GCF (7,56) = 7.
Find the GCF that 7 and 13.
The number 1 is no a element number, yet it is the GCF the 7 and 13, which are both prime numbers.
GCF (7,13) = 1
Let’s briefly return to the cakes. Her mom discovered that:
and that the GCF is 4, for this reason she must divide the cakes right into 4-square-inch pieces. Look in ~ the number that room not component of the GCF:
The yellow numbers tell united state that there are 3 4-square-inch piece in the tiny cake and 5 4-square-inch piece in the big cake. A total of 8 4-square customs pieces renders sense, due to the fact that 8×4=32, which amounts to the cake sizes of 12 and also 20 square inches, because 12+20=32.
Suppose 60 girls and also 48 boys want to pat in a kickball tournament. What is the greatest variety of teams that deserve to be formed that have the same proportion of girl to boys? How numerous girls and also boys would certainly each team have?
3 x 2 x 2 = 12 teams could be formed, every containing 5 girls and 2 x 2 = 4 boys.
The Least typical Multiple
You’re craving hot dogs, so you head to the store and pick up a pack of dogs and also a pack of buns. Typically, a package of franks has 10, while a fill of buns includes 8. This brings us to one of the good mathematical quandaries of our time. After friend eat 8 dogs, you’ll have actually 2 left, however you’ll be out of buns. You’ll need an additional pack that buns for this reason the dogs don’t go to waste. The course, after you eat those, you’ll have actually 6 buns and no dogs. Off to the keep you’ll go, as soon as again, and also the cycle continues.
In bespeak to figure out the hot dog problem, we require to discover the Least typical Multiple (LCM) the buns and dogs. We need the shortest number the is divisible through both the variety of dogs in a pack and also buns in a pack. Once we discover that number, we can figure out how plenty of packs of each to purchase so us don’t have any leftovers.
Let’s start by listing out some multiples the dogs and buns:
There are many typical multiples of dogs and also buns, yet you can see that the LCM of hot dogs and also buns is 40. Other common multiples space 80, 120, etc. To buy 4 package of warm dogs and also 5 packs of buns will certainly ensure a dog for every bun. Hope you’re hungry!
Prime factorization can be used below as well. Recognize the LCM this means is sort of opposing as once finding the GCF. Because that GCF, we desire the common factors. For LCM, we desire the distinctive factors. If a variable occurs in both numbers, we desire the greatest power of that factor. Here goes!
To calculation the LCM, we first need the 5 from factoring 10. Notification that 2 occurs together a aspect of both 10 and also 8. Remember, we need the greatest power of 2:The prime factorization that 10 is \(2\times 5 = 2^1\times 5\). The element factorization the 8 is \(2\times 2\times 2 = 2^3\).
The second number because that our calculation is \(2^3\), because 3 is greater than \(1.5\times 2^3=5\times 8 = 40\), simply as we found previously.
LCM (10,8) = 40
What is the LCM that 15 and also 27?
LCM (15,27) = \(5\times 3^3 = 5\times 27 = 135\).
What is the LCM that 18 and also 36?
The factorizations that both numbers contain 2 and 3 and the highest power the both of castle is 2.
LCM (18,36) = \(2^2\times 3^2 = 4\times 9 = 36\). An alert that 36 is a multiple of 18, also the LCM.
Find the LCM the 7 and also 56.
Again, an alert one of the numbers is prime and it is a variable of the other number. LCM (7,56) = \(7\times 2^3=7\times 8 = 56\).
Find the LCM of 7 and also 13.
The LCM that 7 and 13, both prime numbers, is just LCM(7,13) = 7 x 13 = 91.
Starting in ~ 8:00 a.m., a northbound train stops at a station every 12 minutes and a southbound train stop at the exact same station every 20 minutes. When is the next time both trains will certainly be stopped at the terminal at the exact same time?
LCM (12, 20) = \(2^2\times 3\times 5= 4\times 3\times 5 = 60\). ~ 60 min, or at 9:00, is the next time both trains will certainly be in ~ the terminal at the exact same time.
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There is an east-west train the stops in ~ the terminal every 8 minutes. When is the next time all 3 trains will certainly be quit at the terminal at the exact same time?
LCM (12,20,8) = \(3\times 5\times 2^3 = 120\). After 120 minutes, or at 10:00, is the following time all 3 trains will be in ~ the terminal at the same time.
When trying out with factors and multiples, the GCF and LCM often prove useful. Their an abilities can come in handy for calculations, such together simplifying fractions. However, as we’ve seen, these extend our knowledge of multiplication and also division, pushing united state to truly master what it means for a number to it is in a factor or a multiple, and allow us to make sense of yes, really situations. The best part is, all you have to do to exercise is choose some numbers. We mainly practiced through two at a time, however you can discover the GCF and LCM of any type of group of numbers. Offer it a shot!