Greatest common factor of two multivariate monomials calculator

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.

Please provide numbers separated by a comma "," and click the "Calculate" button to find the GCF.

What is the Greatest Common Factor (GCF)?

In mathematics, the greatest common factor (GCF), also known as the greatest common divisor, of two (or more) non-zero integers a and b, is the largest positive integer by which both integers can be divided. It is commonly denoted as GCF(a, b). For example, GCF(32, 256) = 32.

Prime Factorization Method

There are multiple ways to find the greatest common factor of given integers. One of these involves computing the prime factorizations of each integer, determining which factors they have in common, and multiplying these factors to find the GCD. Refer to the example below.

Prime factorization is only efficient for smaller integer values. Larger values would make the prime factorization of each and the determination of the common factors, far more tedious.

Euclidean Algorithm

Another method used to determine the GCF involves using the Euclidean algorithm. This method is a far more efficient method than the use of prime factorization. The Euclidean algorithm uses a division algorithm combined with the observation that the GCD of two integers can also divide their difference. The algorithm is as follows:

GCF(a, a) = a
GCF(a, b) = GCF(a-b, b), when a > b
GCF(a, b) = GCF(a, b-a), when b > a

In practice:

1. Given two positive integers, a and b, where a is larger than b, subtract the smaller number b from the larger number a, to arrive at the result c.
2. Continue subtracting b from a until the result c is smaller than b.
3. Use b as the new large number, and subtract the final result c, repeating the same process as in Step 2 until the remainder is 0.
4. Once the remainder is 0, the GCF is the remainder from the step preceding the zero result.

From the example above, it can be seen that GCF(268442, 178296) = 2. If more integers were present, the same process would be performed to find the GCF of the subsequent integer and the GCF of the previous two integers. Referring to the previous example, if instead the desired value were GCF(268442, 178296, 66888), after having found that GCF(268442, 178296) is 2, the next step would be to calculate GCF(66888, 2). In this particular case, it is clear that the GCF would also be 2, yielding the result of GCF(268442, 178296, 66888) = 2.

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