What are all the factors, the prime factorization, and factor pairs of 30?

Factors are the building blocks of numbers; they are the integers we can multiply together to get another number. For example, with the number 30 as our focus, understanding its factors provides insight into its mathematical makeup and relationships.

The concept of factors refers to whole numbers that can be multiplied in pairs to reach the target number, which in this case is 30. Factors of 30 therefore include any number that divides 30 without leaving a remainder. Here is the list of factors for the number 30 presented clearly:

• 1
• 2
• 3
• 5
• 6
• 10
• 15
• 30

The most straightforward method to find factors of any number is by division. When dividing 30 by an integer, if the result is also an integer, then both the dividend and divisor are factors. Below is an illustrative example of finding factors through division:

• 30 ÷ 1 = 30, so 1 and 30 are factors of 30.
• 30 ÷ 2 = 15, so 2 and 15 are factors of 30.
• 30 ÷ 3 = 10, so 3 and 10 are factors of 30.
• 30 ÷ 5 = 6, so 5 and 6 are factors of 30.
• Trying to divide 30 by any other integer, such as 4 or 7, will yield a non-integer result, indicating that these numbers are not factors of 30.

Pair factors of a number include two numbers that can be multiplied together to produce that number. For 30, both positive and negative number pairs can be factors:

Prime factorization is the process of breaking down a number into its prime factors. The prime factorization of 30 is found by dividing 30 by the smallest prime number that 30 is divisible by, and continuing the process with the resulting quotients. Below is the factor tree that shows the prime factors of 30:

• 30 ÷ 2 = 15
• 15 ÷ 3 = 5
• 5 is a prime number and cannot be divided further.
• Thus, the prime factorization of 30 = 2 × 3 × 5.

A diagram can be included here to visually demonstrate the factor tree. Note: Graphics are not included in HTML text.

When studying the factors of 30, it’s essential to keep the following points in mind:

• Total number of positive factors of 30 is 8.
• The prime factors of 30 are 2, 3, and 5.
• The pair factors of 30 are (1, 30), (2, 15), (3, 10), and (5, 6).
• The sum of all factors of 30 (1+2+3+5+6+10+15+30) is 72.