There are three ways to find the greatest common factor, how to find the greatest common factor

Use factor-decomposing prime factors.

Method: Divide several numbers into the product of several prime factors, then find the same prime factors, and then multiply these prime factors, and the product is their greatest common factor.

2. Use short division: use short division to divide the array that requires the common factor all the way down until it can no longer be divisible, so that the divisor generated in the short division operation process is the required common factor, the largest of which is the greatest common factor.

Prime factorization, short division, and general division operations.

There are many ways to find the greatest common divisor, the common ones are prime factor factorization, short division, tossing and turning division, and more derogation.

1. Prime factor decomposition.

Prime factor decomposition method: decompose the prime factors of each number separately, and then extract all the common prime factors in each number and multiply, and the resulting product is the greatest common divisor of these numbers.

2. Short division.

To find the greatest common divisor, first use the common divisor of these numbers to continuously divide until all the quotients are co-qualified, and then multiply all the divisors together, and the product obtained is the greatest common divisor of these numbers.

3. Tossing and dividing: Tossing and dividing is a method of finding the greatest common divisor of two natural numbers, also known as Euclidean algorithm.

4. More derogation method: also known as more derogation method, is an algorithm for finding the greatest common divisor from the "Nine Chapters of Arithmetic", which was originally designed for reduction, but it is suitable for any occasion where the greatest common divisor needs to be found.

Generally, we use the first method, for example: find the greatest common divisor of 24 and 60, first decompose the prime factors, and get 24 = 2 2 2 2 3, 60 = 2 2 3 5, all the common prime factors of 24 and 60 are , their product is 2 2 3 = 12, so ) = 12.

Method steps.

1 3 Step by Step Reading.

The first method is enumeration. The so-called enumeration method is to list the factors of two numbers separately, and then find their common factors from them, and finally find the largest common factor from the common factors. For example, find the greatest common factor for .

This method can be used for smaller numbers and is not very convenient for larger numbers.

Factor of 6 ;

Factor of 15 ;

Their common factor is ;

So their greatest common factor is 3.

The second method is short division. Remove both numbers with the prime factor common to these two numbers until the resulting quotient is co-prime (i.e., there is no common factor), and then multiply all the divisors (i.e., the number to the left of the divisor sign), and the product is the greatest common factor of the two numbers. This method is the most concise, the most commonly used, and convenient for the calculation of the greatest common factor for larger numbers.

The third method is to reduce the multiples of these two numbers, first list the factors of the smaller number of these two numbers, and then find out the factors of the larger number from these factors, and find out the common factor of these two numbers, and then find the largest from these common factors, which is the greatest common factor of these two numbers. This method is similar to the first one, but it is not suitable for calculating the greatest common factor of larger numbers.

Precautions. To find the greatest common factor by short division is to multiply the divisor, not by the quotient.

The enumeration method should list all their factors, and there should be no omissions.

1. Enumeration method.

The common factors of 8 and 12 can be listed separately for all the factors of 8 and 12, and then look for them.

Factors of 8: 1, 2, 4, 8.

Factors of 12: 1, 2, 3, 4, 6, 12.

The common factors of 8 and 12 are 1, 2, 4, the largest of which is 4.

You can also find the factor of 8 first, and then find the factor of 12 from the factor of 8.

Factors of 8: 1, 2, 4, 8.

1, 2, 4 are also factors of 12.

The common factors of 8 and 12 are 1, 2, 4, the largest of which is 4.

2. Tossing and dividing (Euclidean algorithm).

Tossing and turning division is to divide the larger of the two numbers by the smaller number, if there is a remainder, then use the smaller number to continue to divide by the remainder, and continue to divide according to this method, until the remainder is 0, then the final divisor is the greatest common factor of the two numbers.

The easiest way to find the greatest common factor is short division. Short division is to write the prime factor common to two numbers where the divisor is written in division, and then drop the quotient of the two numbers divisible by the common prime factor, and then divide, and so on, until the result is co-prime. Finally, multiply all the divisors and the answer is the greatest common factor.

The least common factor is multiplied by the following two answers!

With short division, the left side divides the prime number, and after the division, multiply the numbers in the left row to become the greatest common factor of several numbers.

Common factor: In two or more numbers, if they have the same factor, then this factor(s) is called their common factor. And the one that is the largest of these common factors is called the greatest common factor of these positive integers.

Those numbers can be divided by how many at the same time, until they can't be divided. The product of the numbers divided by the same time is the greatest common factor (simultaneous Ha).

How do you find the common factor? Teach you how to find the greatest common factor, the method is very simple.

Methods and steps for finding the greatest common factor:

1. Write the factor. Write out the respective factors first, then find the common factor, and then find the greatest common factor. This is the most basic method in the new version.

2. With graphics. Write the common factors first, and then write the respective factors.

3. Decompose prime factors. First, decompose the prime factors separately, and then find the common prime factors, if there are more than two, multiply the common prime factors, and the product is the largest common factor; If there is only one, then this prime factor is the greatest common factor of several numbers.

4. Cut-off method. Use the division method to find the greatest common factor of several numbers. Write the numbers first and then divide them with their prime factors until the quotient is co-prime.

If the divisor is one, then this is the greatest common factor of several numbers, and if the divisor is more than two, then the product of the multiplication of the divisor is the greatest common factor of several numbers.

5. Selection. The above four methods can find the greatest common factor of several numbers, but there are advantages and disadvantages to the methods. The first one is easy to understand, but it is cumbersome to do.

The fastest is the division method, so I recommend learning the method of division and the method of decomposing prime factors, so that the efficiency of solving problems will be very high.

Precautions. When using the division method to find the greatest common factor number of several numbers, the quotient must be a coprime number, otherwise the number obtained is not the greatest common factor.

Finding three or more numbers also requires a common factor.

Supplement: The factor common to several numbers is called the common factor of several numbers, and the largest of them is called the greatest common factor. Finding the greatest common factor of two numbers or three numbers is the most common form of primary school, and it is also the most basic knowledge to learn about it in the future.

There are many ways to find the greatest common factor of several numbers, and now we are looking for the fastest method through the study of several methods.

Decomposing prime factor method. Break down several numbers into the product of several prime factors, then find the same prime factors, and then multiply these prime factors, and the product is their greatest common factor.

Take the class seriously, don't skip the class. I won't tell you. Hahahaha.

Decompose two numbers with a common prime number, and finally multiply the prime number.

The greatest factor is the greatest common factor.

Write the factor to find the greatest common factor.

Oh, Jujing can't remember what you're up to.

Hello Bai.

There are two methods for finding the maximum common du factor in primary learning: zhi1, decompose the prime factors of each dao number respectively, and then compare and multiply the prime factors of the inner common capacitance;

2. Using short division, writing short division is similar to the first method, except that the process of finding common factors is merged with the division process.

Short division is difficult to input on a computer, and here we will demonstrate two problems in the first way:

It can be broken down into 2*2*3;32 can be decomposed into 2*2*2*2*2, and the common part is observed to be 2*2. So the greatest common factor of (12,32) is 4.

It can be broken down into 5*3*3*3;25 can be broken down into 5*5, and the common part is observed to be 5. So the greatest common factor of (135,25) is 5.

Generally speaking, to find the greatest common factor of two numbers, the most common way is to find all the factors of these two numbers separately, and then find the common factor of the two numbers, of which the largest one is the greatest common factor of the two numbers, such as finding the greatest common factor of 8 and 12:

Factors of 8: 1, 2, 4, 8.

Factors of 12: 1, 2, 3, 4, 6, 12.

Common factors of 12 and 18: 1, 2, 4

The greatest common factor of 12 and 18: 4

Here are a few quick ways to find the greatest common factor:

1. Multiples method.

When two numbers are multiples, the greatest common factor is the smaller of the two numbers. As.

18 and 9 can directly judge that their greatest common factor is 9, because 18 and 9 are multiples, 9 is the factor of 18, and 9 is also the factor of 9, that is, 9 is the greatest common factor of 18 and 9.

21 and 7 28 and 4 65 and 13

You don't have to think much about the greatest common factor of each group of numbers above, and you can see that they are 7, 4, and 13 in one second.

Second, the mutual quality method.

When two numbers are coprime they are, their greatest common factor is 1. For example, the greatest common factor of 8 and 9 is 1, because the factors of 8 are 1, 2, 4, 8And the factor of 9 has 1,3,9. Then the common factor of 8 and 9 is only 1, which is the greatest common factor.

Therefore, two numbers with only a common factor of 1 are called coprimes, and the greatest common factor of two numbers that are coprime is 1.

Numbers such as 13 and 15 21 and 8 3 and 5 161 and 3 are coprime between each group, so their greatest common factor is 1.

3. Short division.

For two numbers that are not special relations, and the two numbers of the greatest common factor cannot be directly judged, short division can be used. Take two numbers as dividends, and divide them by an identical number (generally not divided by 1, except for o), the number divided by is called the divisor, and the divisor must be able to satisfy the divisibility of two numbers at the same time, in fact, this divisor is the factor of two numbers, and it is divided until it cannot be divided, and then the result of multiplying all the divisors is the greatest common factor of the two numbers.

1. Enumeration method. List the factors of the two numbers separately, and find out that the factor they have in common is their common factor, and the largest one is their greatest common factor.

2. Decomposition of prime factor methodBy decomposing prime factors, you can also easily find the greatest common factor of two numbers.

3. Short divisionShort division is one of the most convenient and commonly used methods for writing, and it is important to guide children to master this method.

Characteristics of the greatest common factor1. The quotient of two numbers divided by their greatest common factor is obtained by mutual quality.

2. The factor of the greatest common factor of two numbers is the factor of these two numbers.

3. If the two numbers are multiples, the smaller number is the greatest common factor of the two numbers.

Enumerative method. For finding the greatest common factor of several smaller positive integers, you can first list all the factors of each positive integer separately, and then find the greatest common factor from their common factors.

Short division. Under the condition that all positive integers are divisible, the prime numbers from smallest to largest are divided by divisors in turn (sometimes the same prime number can be divided several times) until the dividends are mutually primitive, and then the product of multiplying all the divisors is the greatest common factor.

Decomposing prime factor method. According to the properties of the modern mathematical concept of the greatest common factor above4, the standard decomposition of each positive integer to be found can be written separately, and the common prime factor in each decomposition can be written. Each prime factor takes its lowest power in each decomposition equation and multiplies the powers of these prime factors to obtain the greatest common factor.

For example, 24=2x2x2x3, 36=2x2x3x3, after decomposing the prime factors of these two numbers, and multiplying the lowest power of their common prime factors ---2x2x3=12, so that (24,36)=12.

Tossing and dividing. In mathematics, tossing and dividing is also known as Euclidean's algorithm, which is an algorithm for finding the greatest common factor. Tossing and dividing first appeared in Euclid's Geometry in 300 BC, while in the Eastern Han Dynasty, it can be traced back to the Nine Chapters of Arithmetic.

The greatest common factor of two positive integers is the largest positive integer that can be divisible by both of them. Tossing division is based on the principle that the greatest common factor of two positive integers is equal to the greatest common factor of the smaller of them and the difference between the two numbers.

For example, the greatest common factor of 252 and 105 is 21 (252 = 21 12, 105 = 21 5), and since 252-105 = 147, the greatest common factor of 147 and 105 is also 21. In the process, the larger number shrinks, so continuing the same calculation will keep shrinking the two numbers until one of them becomes zero. At this point, the remaining number that has not yet become zero is the greatest common factor of the two numbers.