Find the greatest common monomial factor calculator

Explanation:

#12x^2 = 2*2*3*x*x#

#8x^3 = 2*2*2*x*x*x#

For H.C.F., take the factors which are common in both.

So, #the H.C.F. = 2*2*x*x#
=# 4x^2#

You can say that #4x^2# is the greatest common monomial factor of #8x^3 and 12x^2#.

Video transcript

Find the greatest common factor of these monomials. Now, the greatest common factor of anything is the largest factor that's divisible into both. If we're talking about just pure numbers, into both numbers, or in this case, into both monomials. Now, we have to be a little bit careful when we talk about greatest in the context of algebraic expressions like this. Because it's greatest from the point of view that it includes the most factors of each of these monomials. It's not necessarily the greatest possible number because maybe some of these variables could take on negative values, maybe they're taking on values less than 1. So if you square it, it's actually going to become a smaller number. But I think without getting too much into the weeds there, I think if we just kind of run through the process of it, you'll understand it a little bit better. So to find the greatest common factor, let's just essentially break down each of these numbers into what we could call their prime factorization. But it's kind of a combination of the prime factorization of the numeric parts of the number, plus essentially the factorization of the variable parts. If we were to write 10cd squared, we can rewrite that as the product of the prime factors of 10. The prime factorization of 10 is just 2 times 5. Those are both prime numbers. So 10 can be broken down as 2 times 5. c can only be broken down by c. We don't know anything else that c can be broken into. So 2 times 5 times c. But then the d squared can be rewritten as d times d. This is what I mean by writing this monomial essentially as the product of its constituents. For the numeric part of it, it's the constituents of the prime factors. And for the rest of it, we're just kind of expanding out the exponents. Now, let's do that for 25c to the third d squared. So 25 right here, that's 5 times 5. So this is equal to 5 times 5. And then c to the third, that's times c times c times c. And then d squared, times d squared. d squared is times d times d. So what's their greatest common factor in this context? Well, they both have at least one 5. Then they both have at least one c over here. So let's just take up one of the c's right over there. And then they both have two d's. So the greatest common factor in this context, the greatest common factor of these two monomials is going to be the factors that they have in common. So it's going to be equal to this 5 times-- we only have one c in common, times-- and we have two d's in common, times d times d. So this is equal to 5cd squared. And so 5d squared, we can kind of view it as the greatest. But I'll put that in quotes depending on whether c is negative or positive and d is greater than or less than 0. But this is the greatest common factor of these two monomials. It's divisible into both of them, and it uses the most factors possible.

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Calculator Use

Calculate GCF, GCD and HCF of a set of two or more numbers and see the work using factorization.

Enter 2 or more whole numbers separated by commas or spaces.

The Greatest Common Factor Calculator solution also works as a solution for finding:

• Greatest common factor (GCF)
• Greatest common denominator (GCD)
• Highest common factor (HCF)
• Greatest common divisor (GCD)

What is the Greatest Common Factor?

The greatest common factor (GCF or GCD or HCF) of a set of whole numbers is the largest positive integer that divides evenly into all numbers with zero remainder. For example, for the set of numbers 18, 30 and 42 the GCF = 6.

Greatest Common Factor of 0

Any non zero whole number times 0 equals 0 so it is true that every non zero whole number is a factor of 0.

k × 0 = 0 so, 0 ÷ k = 0 for any whole number k.

For example, 5 × 0 = 0 so it is true that 0 ÷ 5 = 0. In this example, 5 and 0 are factors of 0.

GCF(5,0) = 5 and more generally GCF(k,0) = k for any whole number k.

However, GCF(0, 0) is undefined.

How to Find the Greatest Common Factor (GCF)

There are several ways to find the greatest common factor of numbers. The most efficient method you use depends on how many numbers you have, how large they are and what you will do with the result.

Factoring

To find the GCF by factoring, list out all of the factors of each number or find them with a Factors Calculator. The whole number factors are numbers that divide evenly into the number with zero remainder. Given the list of common factors for each number, the GCF is the largest number common to each list.

Example: Find the GCF of 18 and 27

The factors of 18 are 1, 2, 3, 6, 9, 18.

The factors of 27 are 1, 3, 9, 27.

The common factors of 18 and 27 are 1, 3 and 9.

The greatest common factor of 18 and 27 is 9.

Example: Find the GCF of 20, 50 and 120

The factors of 20 are 1, 2, 4, 5, 10, 20.

The factors of 50 are 1, 2, 5, 10, 25, 50.

The factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.

The common factors of 20, 50 and 120 are 1, 2, 5 and 10. (Include only the factors common to all three numbers.)

The greatest common factor of 20, 50 and 120 is 10.

Prime Factorization

To find the GCF by prime factorization, list out all of the prime factors of each number or find them with a Prime Factors Calculator. List the prime factors that are common to each of the original numbers. Include the highest number of occurrences of each prime factor that is common to each original number. Multiply these together to get the GCF.

You will see that as numbers get larger the prime factorization method may be easier than straight factoring.

Example: Find the GCF (18, 27)

The prime factorization of 18 is 2 x 3 x 3 = 18.

The prime factorization of 27 is 3 x 3 x 3 = 27.

The occurrences of common prime factors of 18 and 27 are 3 and 3.

So the greatest common factor of 18 and 27 is 3 x 3 = 9.

Example: Find the GCF (20, 50, 120)

The prime factorization of 20 is 2 x 2 x 5 = 20.

The prime factorization of 50 is 2 x 5 x 5 = 50.

The prime factorization of 120 is 2 x 2 x 2 x 3 x 5 = 120.

The occurrences of common prime factors of 20, 50 and 120 are 2 and 5.

So the greatest common factor of 20, 50 and 120 is 2 x 5 = 10.

Euclid's Algorithm

What do you do if you want to find the GCF of more than two very large numbers such as 182664, 154875 and 137688? It's easy if you have a Factoring Calculator or a Prime Factorization Calculator or even the GCF calculator shown above. But if you need to do the factorization by hand it will be a lot of work.

How to Find the GCF Using Euclid's Algorithm

1. Given two whole numbers, subtract the smaller number from the larger number and note the result.
2. Repeat the process subtracting the smaller number from the result until the result is smaller than the original small number.
3. Use the original small number as the new larger number. Subtract the result from Step 2 from the new larger number.
4. Repeat the process for every new larger number and smaller number until you reach zero.
5. When you reach zero, go back one calculation: the GCF is the number you found just before the zero result.

For additional information see our Euclid's Algorithm Calculator.

Example: Find the GCF (18, 27)

27 - 18 = 9

18 - 9 - 9 = 0

So, the greatest common factor of 18 and 27 is 9, the smallest result we had before we reached 0.

Example: Find the GCF (20, 50, 120)

Note that the GCF (x,y,z) = GCF (GCF (x,y),z). In other words, the GCF of 3 or more numbers can be found by finding the GCF of 2 numbers and using the result along with the next number to find the GCF and so on.

Let's get the GCF (120,50) first

120 - 50 - 50 = 120 - (50 * 2) = 20

50 - 20 - 20 = 50 - (20 * 2) = 10

20 - 10 - 10 = 20 - (10 * 2) = 0

So, the greatest common factor of 120 and 50 is 10.

Now let's find the GCF of our third value, 20, and our result, 10. GCF (20,10)

20 - 10 - 10 = 20 - (10 * 2) = 0

So, the greatest common factor of 20 and 10 is 10.

Therefore, the greatest common factor of 120, 50 and 20 is 10.

Example: Find the GCF (182664, 154875, 137688) or GCF (GCF(182664, 154875), 137688)

First we find the GCF (182664, 154875)

182664 - (154875 * 1) = 27789

154875 - (27789 * 5) = 15930

27789 - (15930 * 1) = 11859

15930 - (11859 * 1) = 4071

11859 - (4071 * 2) = 3717

4071 - (3717 * 1) = 354

3717 - (354 * 10) = 177

354 - (177 * 2) = 0

So, the the greatest common factor of 182664 and 154875 is 177.

Now we find the GCF (177, 137688)

137688 - (177 * 777) = 159

177 - (159 * 1) = 18

159 - (18 * 8) = 15

18 - (15 * 1) = 3

15 - (3 * 5) = 0

So, the greatest common factor of 177 and 137688 is 3.

Therefore, the greatest common factor of 182664, 154875 and 137688 is 3.

References

[1] Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae, 31st Edition. New York, NY: CRC Press, 2003 p. 101.

[2] Weisstein, Eric W. "Greatest Common Divisor." From MathWorld--A Wolfram Web Resource.

Help With Fractions: Finding the Greatest Common Factor.

Wikipedia: Euclidean Algorithm.

How do you find the greatest common monomial factor?

What is the greatest common monomial factor of 12x 2 and 8x 3?

How do you factor a monomial step by step?