What is the difference between common divisor and common factor?

1, different definitions

Common divisor: Common divisor, also called "common factor". Is an integer divisible by several integers at the same time. If an integer is a divisor of several integers, it is called their "common divisor"; The greatest common divisor is called the greatest common divisor For any number of positive integers, 1 is always their common factor.

Common factor: Common factor is a mathematical concept, which means that an integer can be divided by several integers at the same time and can be calculated by division.

2. Different concepts

Common divisor: If the number A is divisible by the number B, then A is called a multiple of B, and B is called a divisor of A. Both divisor and multiple represent the relationship between an integer and another integer and cannot exist independently. If we can only say that 16 is a multiple of a certain number and 2 is a divisor of a certain number, then we cannot say that 16 is a multiple and 2 is a divisor in isolation.

"Multiply" and "multiple" are two different concepts. "Times" refers to the quotient of dividing two numbers, which can be integer, decimal or fraction. "Multiplication" is only the concept of a number within the divisible range of a number, which is relative to "divisor" and represents a number that can be divisible by natural numbers.

Several integers, common divisors, are called the common divisors of these numbers; The largest one is called the greatest common divisor of these numbers. For example, the common divisors of 12 and 16 are 1, 2 and 4, the largest of which is 4, which is the greatest common divisor of 12 and 16, generally denoted as (12,/kloc. The greatest common divisor of 12, 15 and 18 is 3, and it is (12, 15, 18)=3.

Common factor: let a and b be two integers. If c is an integer and is divisible by a, then c is called a factor (or divisor) of a, all divisors of a constitute a nonempty set (set A), and all factors of b constitute a set B, let.

The element of C is the common factor of A and B. Obviously, C is not empty, because at least

. A common factor is an integer that can be divisible by several integers at the same time. For example, all common factors of 4 and 6 are 1, 2,-1, 2, and all common factors appear in pairs in the form of opposites, so it is enough to generally study positive factors, so the common factors of 4 and 6 are 1, 2.

3, the solution is different.

Common divisor: a method to find the greatest common divisor of two numbers

Multiple relations

If the larger number is a multiple of the smaller number, then the smaller number is the greatest common divisor of these two numbers.

Co-prime relation

If these two numbers are prime numbers, their greatest common divisor is 1.

Common factor: 54 can be expressed as the product of two different positive integers:

So the positive divisors of 54 are 1, 2, 3, 6, 9, 18, 27, 54.

Similarly, 24 can be expressed as:

So the positive divisor of 24 is 1, 2, 3, 4, 6, 8, 12, 24.

The positive divisor 1 with both 24 and 54, 2, 3 and 6 are common divisors.

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