Divisor

A divisor is a special kind of number in mathematics. It is an integer that you can multiply by another integer to get a specific number. For example, 2 is a divisor of 4 because 2 multiplied by 2 equals 4. In this case, we also say that 4 is a multiple of 2.

When we talk about divisors, we are often interested in whether one number can divide another evenly. This means that when you divide the larger number by the smaller one, there is no remainder left over. For instance, 3 is a divisor of 6 because 6 divided by 3 equals 2 with no remainder.

Divisors are important in many areas of math. They help us understand factors and multiples, which are key ideas in arithmetic and number theory. Knowing about divisors can also help with problems involving fractions, simplifying numbers, and finding common factors between different numbers.

In short, a divisor is a number that fits perfectly into another number. It plays a big role in how we work with integers and solve many types of math problems. For more information about how divisors relate to division, you can read about Division (mathematics). There are also other meanings for the word "divisor," which you can explore in the article Divisor (disambiguation).

Definition

An integer can be divided evenly by another integer if you can multiply that second integer by another integer to get the first one. For example, 6 can be divided by 2 because 2 × 3 = 6. In this case, we say that 2 is a divisor or factor of 6, and 6 is a multiple of 2.

Mathematicians write this relationship using a special symbol: if 2 is a divisor of 6, they write it as 2 ∣ 6. If 2 is not a divisor of 6, they write 2 ∤ 6. This helps us clearly show whether one number divides another without any remainder.

General

Divisors are numbers that can multiply with another number to make a given integer. For example, the divisors of 4 are 1, 2, and 4 because 1 × 4 = 4, 2 × 2 = 4, and 4 × 1 = 4. Sometimes negative numbers like -1, -2, and -4 can also be divisors, but we usually focus on positive ones.

Numbers that can be divided by 2 are called even, while those that cannot are called odd. Every number is a divisor of itself, and 1 and -1 are divisors of every number. Numbers with divisors other than 1 and themselves are called composite numbers, while prime numbers only have 1 and themselves as divisors. There are also special rules called divisibility rules that help us find divisors by looking at the digits of a number.

Examples

A divisor of a number is another number that can be multiplied to give the first number. For example, 7 is a divisor of 42 because 7 times 6 equals 42. We can also say that 42 is divisible by 7, or that 42 is a multiple of 7.

The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42. This means each of these numbers can multiply with another number to equal 42.

Further notions and facts

Divisibility in mathematics describes how one number can be divided evenly by another. If a number a divides another number b evenly, we say a is a divisor of b. This means b can be multiplied by some whole number to equal a.

Some important rules include:

• If a divides b and b divides c, then a also divides c.
• If a divides b and b divides a, then a and b are the same or opposites.
• A number that only has 1 and itself as divisors is called a prime number.

For example, the divisors of 6 are 1, 2, 3, and 6. A proper divisor is any divisor of a number except the number itself, so for 6, the proper divisors are 1, 2, and 3.