For example, for integers 12 and 18, their greatest common factor is 6, because 6 is the largest positive integer that can divide 12 and 18 simultaneously.
Solution of the greatest common factor
There are many ways to find the greatest common factor, and the common methods are prime factor decomposition and Euclid algorithm. No matter which method is adopted, the final result is to find the greatest common divisor in this group of numbers. The greatest common factor is often used in mathematics and computer science to simplify fractions, reduce proportions and solve congruence equations.
There are several common methods to find the greatest common factor (GCD):
1. prime factorization method
Divide two or more numbers by prime factors respectively, then find out all their common prime factors, multiply these common prime factors, and the product is the greatest common factor.
2. Commutation division (Euclid algorithm)
Take the larger number of two numbers and divide it by the smaller number to get the quotient and the remainder. Then divide the smaller number by the remainder, and then get the quotient and the new remainder. Repeat this process until the remainder is 0, at which point the divisor is the greatest common factor.
3. More phase subtraction
Take the larger one of the two numbers and subtract the smaller one to get the difference. Then subtract a smaller number from the difference to get a new difference. Repeat this process until the difference is 0 or two numbers are equal, at which time this number is the greatest common factor.
4. The combination of tossing and turning subtraction and shifting (more efficient Euclidean algorithm)
On the basis of subtraction, shift operation is introduced to speed up the calculation process.
Application of the greatest common factor
1. Reduced score
When a fraction needs to be reduced, the numerator and denominator can be reduced by the greatest common factor. By dividing the numerator and denominator by their greatest common factor, you can get the reduced score and keep it in the simplest form.
2. Solve the problem of modular operation
In modular operation, it is necessary to solve congruence equation. The greatest common factor plays a key role in judging whether two numbers are coprime and solving modular linear equations.
3. Decomposition polynomial
In algebra, the greatest common factor is used to decompose polynomials. By finding the greatest common factor of polynomials, polynomials can be decomposed into smaller factor products, thus simplifying the calculation and analysis process.
4. RSA algorithm in cryptography
RSA algorithm is a commonly used public key encryption and digital signature algorithm, and one of the key steps is to solve the greatest common factor of two large prime numbers to ensure security and reliability.
5. Algorithm design and optimization
The greatest common factor algorithm also plays an important role in algorithm design and optimization. For example, some sorting algorithms use the greatest common factor to realize the cyclic shift operation, thus improving the execution efficiency.
An example of finding the greatest common factor
Question: Find the greatest common factor of integers 24 and 36.
Solution: It can be solved by the division of phases. First, divide 36 by 24 to get the quotient 1 and the remainder 12. Then, divide 24 by 12 to get quotient 2 and remainder 0. At this point, the remainder is 0, so the greatest common factor is the divisor of the previous step, which is 12. So the greatest common factor of 24 and 36 is 12.
In actual calculation, you can also use the prime factor decomposition method to decompose 24 and 36 into prime factors respectively:
24 = 2^3 * 3
36 = 2^2 * 3^2
It can be seen that their common prime factor is the square of 2 and the first power of 3. Multiply these common prime factors to get 2 2 * 3 = 12, that is, the maximum common factor is12.
No matter which method you use, you can finally get the same result, that is, the greatest common factor of 24 and 36 is 12.