Gcf Family Tree of 54 and 42 What Is the Greatest Common Factor of 6 and 5

The GCF calculator evaluates the Greatest Common Factor between two to 6 different numbers. Read on to find the respond to the question: "What is the Greatest Common Factor of given numbers?", learn about several GCF finder methods, including prime factorization or the Euclidean algorithm, decide which is your favorite, and check out by yourself that our GCF calculator can salvage you lot time when dealing with large numbers!

What is GCF?

The Greatest Common Factor definition is the largest integer factor that is present between a gear up of numbers. Information technology is also known as the Greatest Mutual Divisor, Greatest Common Denominator (GCD), Highest Common Factor (HCF), or Highest Mutual Divisor (HCD). This is important in certain applications of mathematics such as simplifying polynomials where often it'due south essential to pull out common factors. Next, we need to know how to detect the GCF.

How to Find the Greatest Mutual Factor

There are various methods which help you to detect GCF. Some of them are child's play, while others are more than complex. It's worth knowing all of them so you lot can decide which you prefer:

• Using the list of factors,
• Prime factorization of numbers,
• Euclidean algorithm,
• Binary algorithm (Stein'due south algorithm),
• Using multiple backdrop of GCF (including Least Mutual Multiple, LCM).

The good news is that you can approximate the GCD with uncomplicated math operations, without roots or logarithms! For most cases they are just subtraction, multiplication, or division.

GCF finder - list of factors

The main method used to estimate the Greatest Mutual Divisor is to find all of the factors of the given numbers. Factors are merely numbers which multiplied together effect in the original value. In general, they can be both positive and negative, e.g. 2 * 3 is the same every bit (-2) * (-iii), both equal 6. From a applied point of view, we consider simply positive ones. Moreover, merely integers are concerned. Otherwise, you cound find an space combination of distinct fractions being factors, which is pointless in our case. Knowing that, let's approximate the Greatest Common Denominator of numbers 72 and twoscore.

1. Factors of 72 are: ane, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72,
2. Factors of twoscore are: 1, 2, 4, 5, 8, 10, 20, 40,
3. List all the common factors: 1, 2, four, 8,
4. The Greatest Common Divisor is 8, the highest value from above.

Lets endeavor something more challenging. We want to find the respond for a question: "What is the Greatest Common Gene of 33264 and 35640?" All we need to do is repeat the previous steps:

1. Factors of 33264 are : 1, two, 3, four, 6, seven, 8, 9, xi, 12, 14, xvi, xviii, 21, 22, 24, 27, 28, 33, 36, 42, 44, 48, 54, 56, 63, 66, 72, 77, 84, 88, 99, 108, 112, 126, 132, 144, 154, 168, 176, 189, 198, 216, 231, 252, 264, 297, 308, 336, 378, 396, 432, 462, 504, 528, 594, 616, 693, 756, 792, 924, 1008, 1188, 1232, 1386, 1512, 1584, 1848, 2079, 2376, 2772, 3024, 3696, 4158, 4752, 5544, 8316, 11088, 16632, 33264,
2. Factors of 35640 are: ane, 2, 3, 4, 5, 6, viii, nine, ten, 11, 12, 15, 18, 20, 22, 24, 27, thirty, 33, 36, twoscore, 44, 45, 54, 55, 60, 66, 72, 81, 88, 90, 99, 108, 110, 120, 132, 135, 162, 165, 180, 198, 216, 220, 264, 270, 297, 324, 330, 360, 396, 405, 440, 495, 540, 594, 648, 660, 792, 810, 891, 990, 1080, 1188, 1320, 1485, 1620, 1782, 1980, 2376, 2970, 3240, 3564, 3960, 4455, 5940, 7128, 8910, 11880, 17820, 35640,
3. List of all common divisors: one, ii, three, 4, 6, viii, 9, xi, 12, 18, 22, 24, 27, 33, 36, 44, 54, 66, 72, 88, 99, 108, 132, 198, 216, 264, 297, 396, 594, 792, 1188, 2376,
4. The final consequence is: 2376.

Every bit you tin can see, the higher the number of factors, the more than fourth dimension consuming the procedure gets, and it's easy to make a mistake. It's worth knowing how this method works, but instead, we recommend to use our GCF figurer, just to make sure that the outcome is correct.

Prime factorization

Another commonly used procedure which can be treated as a Greatest Common Divisor reckoner utilizes the prime number factorization. This method is somewhat related to the one previously mentioned. Instead of listing all of the possible factors, we find simply the ones which are prime numbers. As a result, the production of all shared prime numbers is the answer to our problem, and what'south more than important, there is always one unique style to factorize any number to prime ones. So now, permit's find the Greatest Common Denominator of 72 and 40 using prime factorization:

1. Prime factors of 72 are: two, ii, two, 3, 3,
2. Prime factors of 40 are: 2, 2, 2, 5,
3. In other words, we can write: 72 = two * 2 * 2 * three * 3 and xl = two * ii * 2 * 5,
4. The part which is shared in both cases is 2 * two * 2 = viii, and that'south the Greatest Common Gene.

We can see that for this simple example the result is consistent with the previous method. Let's detect if it works equally well for the more complicated example. What is the GCF of 33264and 35640?

1. Prime factors of 33264 are: 2, 2, ii, two, 3, 3, iii, 7, 11,
2. Prime factors of 35640 are: 2, 2, 2, iii, 3, 3, 3, 5, 11,
3. We tin can utilise exponent notation to write products as: 33264 = 2⁴ * 3³ * 7 * 11, 35640 = two³ * 3⁴ * 5 * xi,
4. The common product of 2 numbers is 2³ * 3³ * 11. We can also write it in a more compact and sophisticated way, with factorials taken into account: (3!)³ * 11. Check out if our GCD calculator gives yous the same consequence, which is 2376.

Euclidean algorithm

The idea which is the basis of the Euclidean algorithm says that if the number k is the Greatest Common Cistron of numbers A and B, then k is also GCF for the departure of these numbers A - B. Following this procedure, we will finally achieve 0. Equally a effect, the Greatest Common Divisor is the concluding nonzero number. Let'southward take a await at our examples 1 more time - numbers twoscore and 72. Each time we make a subtraction we compare two numbers, ordering them from the highest to the smallest value:

• GCF of 72 and twoscore: a deviation 72 - 40 equals 32,
• GCF of 40 and 32: xl - 32 = viii,
• GCF of 32 and 8: 32 - 8 = 24,
• GCF of 24 and 8: 24 - 8 = 16,
• GCF of 16 and eight: 16 - eight = viii,
• GCF of 8 and 8: 8 - 8 = 0 Finish!

In our final footstep, we obtain 0 from subtraction. This ways that we notice our Greatest Common Divisor and its value in the penultimate line of the subtractions: 8.

What near more hard example with 33264 and 35640? Let's endeavor to solve information technology using Euclidean algorithm:

• GCF of 35640 and 33264: 35640 - 33264 = 2376,
• GCF of 33264 and 2376: 33264 - 2376 = 30888,
• GCF of 30888 and 2376: 30888 - 2376 = 28512,
• GCF of 28512 and 2376: 28512 - 2376 = 26136,
• GCF of 26136 and 2376: 26136 - 2376 = 23760,
• GCF of 23760 and 2376: 23760 - 2376 = 21384,
• GCF of 21384 and 2376: 21384 - 2376 = 19008,
• GCF of 19008 and 2376: 19008 - 2376 = 16632,
• GCF of 16632 and 2376: 16632 - 2376 = 14256,
• GCF of 14256 and 2376: 14256 - 2376 = 11880,
• GCF of 11880 and 2376: 11880 - 2376 = 9504,
• GCF of 9504 and 2376: 9504 - 2376 = 7128,
• GCF of 7128 and 2376: 7128 - 2376 = 4752,
• GCF of 4752 and 2376: 4752 - 2376 = 2376,
• GCF of 2376 and 2376: 2376 - 2376 = 0 STOP!

Similarly to the previous example, the GCD of 33264 and 35640 is the last nonzero divergence in the procedure, which is 2376.

As y'all can see, the basic version of this GCF finder is very efficient and straightforward but has one significant drawback. The bigger the divergence between the given numbers, the more than steps are needed to reach the concluding step. The modulo is an effective mathematical operation which solves the result considering we are interested only in the rest smaller than both numbers. Permit'southward repeat the Euclidean algorithm for our examples using modulo instead of ordinary subtraction:

• GCF of 72 and 40: 72 mod 40 = 32,
• GCF of 40 and 32: 40 modern 32 = eight,
• GCF of 32 and 8: 32 modern eight = 0 STOP!

The Greatest Common Denominator is 8. What virtually the other one?

• GCF of 35640 and 33264: 35640 mod 33264 = 2376,
• GCF of 33264 and 2376: 33264 mod 2376 = 0 STOP!

GCD of 35640 and 33264 is 2376, and information technology's found in just two steps instead of fifteen. Not bad, is it?

Binary Greatest Common Divisor algorithm

If yous like arithmetics operations simpler than those used in the Euclidean algorithm (e.k. modulo), the Binary algorithm (or Stein's algorithm) is definitely for you! All you have to use is comparison, subtraction, and division past 2. While estimating the Greatest Common Factor of ii numbers, keep in mind these identities:

1. gcd(A, 0) = A, we are using the fact that each number divides zero and an observation from the last step in Euclidean algorithm - ane of the numbers drib to nix, and our result was the previous one,
2. If both A and B are even it ways that gcd(A, B) = 2 * gcd(A/2, B/2), due to the fact that two is a common factor,
3. If only one of the numbers is even, allow's say A, than gcd(A, B) = gcd(A/two, B). This time 2 is not a common divisor then nosotros tin can continue with the reduction until both numbers are odd,
4. If both A and B are odd and A > B, then gcd(A, B) = gcd((A-B)/ii, B). This time we combine 2 features into one footstep. The first one is derived from the Euclidean algorithm, working out the Greatest Mutual Divisor of the departure of both numbers and the smaller one. Secondly, the division by 2 is possible since the departure of 2 odd numbers is even, and according to pace 3 we can reduce the fifty-fifty one.
5. Steps 2-4 are repeated until reaching step 1 or if A = B. The event volition be 2ⁿ * A, where n is the number of factors 2 found in a second step.

As usual, allow's practice the algorithm with our sets of numbers. Nosotros offset with 40 and 72:

• They are both withal gcf(72, 40) = 2 * gcf(36, xx) = 2² * gcf(eighteen, 10) = two³ * gcf(9, 5) = …,
• The remaining numbers are odd so … = 2³ * gcf((ix-5)/2, five) = ii³ * gcf(ii, 5),
• 2 is even so we can reduce information technology: … = 2³ * gcf(i, 5),
• 1 and 5 are odd and so: … = 2³ * gcf((5-1)/2, 1) = 2³ * gcf(2, 1),
• Remove two from an even number: … = 2³ * gcf(1, 1) = ii³ = 8.

Actually, nosotros could've stopped at the 3rd pace since GCD of 1 and any number is 1.
Okay, and how to find the Greatest Mutual Factor of 33264 and 35640 using the binary method?

• Two even numbers: gcf(35640, 33264) = ii* gcf(17820, 16632) = 2² * gcf(8910, 8316) = 2³ * gcf(4455, 4158) = …,
• I fifty-fifty one odd: … = 2³ * gcf(4455, 2079),
• 2 odd: … = 2³ * gcf((4455-2079)/ii, 2079) = 2³ * gcf(1188, 2079),
• One even one odd: … = 2³ * gcf(594, 2079) = 2³ * gcf(297, 2079),
• Two odd: … = 2³ * gcf((2079-297)/ii, 297) = 2³ * gcf(891, 297),
• Two odd: … = ii³ * gcf((891-297)/two, 297) = two³ * gcf(297, 297) = ii³ * 297 = 2376.

Coprime numbers

We know that prime number numbers are those that have but two positive integer factors: one and itself. So the question is, what are coprime numbers? We can ascertain them as numbers which have no common factors. More precisely, 1 is their only common factor, only since nosotros omit 1 in prime factorization, it'south okay to say that they have no common divisors. In other words, we can write that numbers A and B are coprime if gcf(A,B) = 1. It doesn't actually hateful that either of them is a prime number, just the list of shared factors is empty. The examples of coprime numbers are: v and 7, 35 and 48, 23156 and 44613.

A fun fact: it's possible to calculate the probability that two randomly chosen numbers are coprime. Although information technology's quite complicated, the overall result is about 61%. Are you surprised? Just exam information technology by yourself - imagine two random numbers (let's say of at to the lowest degree v digits), use our Greatest Mutual Factor calculator and find if the consequence is 1 or not. Repeat the game multiple times and guess what's the pct of coprime numbers you plant.

Greatest Mutual Denominator of more 2 numbers

At present that we are enlightened of numerous methods of finding the Greatest Mutual Divisor of 2 numbers, you might ask: "how to detect the Greatest Mutual Factor of three or more numbers?". Information technology turns out not to exist as difficult as it might seem at outset glance. Well, listing all of the factors for each number is definitely a straightforward method because we can only observe the greatest 1. However, you tin can apace realize that it gets more and more than time consuming equally the number of figures increases.

Prime factorization method has a similar drawback, only since we tin can group all of the primes in, for example, ascending order, we tin introduce a style to work out a result a little faster than previously.

On the other mitt, if you prefer using binary or Euclidean algorithms to estimate what is the GCF of multiple numbers, you can as well use a theorem which states that:

gcf(a, b, c) = gcf(gcf(a, b), c) = gcf(gcf(a, c), b) = gcf(gcf(b, c), a).

It ways that nosotros tin calculate the GCD of whatever two numbers and and then start the algorithm over again using the upshot and the third number, and go on as long as there are whatsoever figures left. It doesn't thing which ii we choose get-go.

Least Mutual Multiple

Another concept closely related to GCD is the To the lowest degree Common Multiple. To find the Least Common Multiple, nosotros utilize much of aforementioned process we used to find the GCF. In one case nosotros get the numbers downwards to the prime factorization, we look for the smallest power of each cistron, as opposed to the largest power. And then we multiply the highest powers, and the result is the Least Common Multiple or LCM. This can exist washed past hand or with the use of the LCM calculator.

Greatest Common Cistron can be estimated with the employ of LCM. The following expression is valid:

gcf(a, b) = |a * b| / lcm(a, b).

It may be handy to find the Least Common Multiple first, due to the complexity and duration. Naturally, it tin be calculated either fashion, and then it's worth knowing both how to find GCD and LCM.

Backdrop of GCD

Nosotros have already presented few backdrop of Greatest Mutual Denominator. In this department, nosotros list the most important ones:

• If the ratio of two numbers a and b (a > b) is an integer then gcf(a, b) = b. (If y'all're in doubt what'southward the ratio of these two numbers, yous tin ever use our ratio calculator!),

• gcf(a, 0) = a, used in Euclidean algorithm,

• gcf(a, 1) = 1,

• If a and b don't accept common factors (they are coprime) then gcf(a, b) = 1,

• All common factors of a and b are also divisors of gcf(a,b),

• If b * c / a is an integer and gcf(a, b) = d, then a * c / d is likewise an integer,

• For any integer k: gcf(chiliad*a, k*b) = 1000 * gcf(a, b), used in binary algorithm,

• For whatsoever positive integer chiliad: gcf(a/yard, b/k) = gcf(a, b) / chiliad,

• gcf(a, b) * lcm(a, b) = |a*b|,

• gcf(a, lcm(b, c)) = lcm(gcf(a, b), gcf(a, c)),

• lcm(a, gcf(b, c)) = gcf(lcm(a, b), lcm(a, c)).

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Source: https://www.omnicalculator.com/math/gcf

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