A factor is the collective name for the two parts of a multiplication: the multiplier and the multiplicand. When a number is presented as a multiplication, it is said to be divided into factors. The division into factors can be continued up to the number's initial factors. When this happens, the number is presented as the product of prime numbers.

A prime number is a number that is divisible only by [[$ 1 $]] and by itself. Thus, the prime number itself has exactly two factors. If a number is not a prime number, it is considered a combined number. Each integer [[$ (≥2) $]] can be represented as the product of prime numbers in only a single way. Generally, the prime factors are ordered in order of magnitude, and the same prime factors are grouped together into a power.

Example 1

Divide the number [[$ 140 $]] into factors and further into prime factors.

[[$ \begin{align*} 140 &= 14 \cdot 10 \;\;\;\;\;\;\;\;\; {\color {blue} {\text {divided into factors}}} \\ &= 2 \cdot 7 \cdot2 \cdot 5 \\ &= 2^2 \cdot 5 \cdot 7 \;\;\;\;\;\; {\color {red} {\text {divided into primary factors}}} \\ \end{align*} $]]​

Example 2

Determine the largest common factor of numbers [[$ 140 $]] and [[$ 180 $]], as well as their smallest common dividend.

First, divide the numbers into their prime factors.

[[$ 140 = 14 \cdot 10 = 2 \cdot 7 \cdot 5 = 2^2 \cdot 5 \cdot 7 \\ 180 = 18 \cdot 10 = 2 \cdot 9 \cdot 2 \cdot 5 = 2 \cdot 3 \cdot 3 \cdot 2 \cdot5 = 2^2 \cdot 3^2 \cdot 5 \\ \\ \\ {\color {red} {\text {The biggest common factor is}}} \;\;\;\; 2^2 \cdot 5 = 20 \\ {\color {blue} {\text {The smallest common dividend is}}} \;\;\;\; 2^2 \cdot 3^2 \cdot 5 \cdot 7 \\ $]]​

Note! In the smallest common dividend, common factors are considered only once.

By dividing the denominator and numerator of a fraction by prime factors, we see what kind of a decimal number it is. The ending decimal number is obtained when the denominator of the fraction in its simplified form is only two or five. If the denominator begins with other numbers, it is an infinite periodic decimal number.

Example 3

Let's study with the help of prime factors what kind of decimal numbers the following values are.

a) [[$ \displaystyle\frac {3} {30} = \displaystyle\frac {3} {2 \cdot 3 \cdot 5 } = \displaystyle\frac {1} {2 \cdot 5 } \;\;\; {\color {blue} {\text {The denominator only contains the numbers 2 and 5, so it is an ending decimal number.}}} $]]​

b) ​[[$ \displaystyle\frac {5} {30} = \displaystyle\frac {5} {2 \cdot 3 \cdot 5 } = \displaystyle\frac {1} {2 \cdot 3 } \;\;\; {\color {blue} {\text {An infinite periodic decimal number because the denominator has a number 3.}}} $]]​