What are all the factors, the prime factorization, and factor pairs of 27?

Factors are the numbers we multiply together to get another number. Understanding these factors sheds light on the structure and properties of a number within mathematics. Taking 27 as our illustrative example, factors of a number like 27 are integral to a plethora of mathematical concepts, including division, prime factorization, and the analysis of number patterns.

The concept of factors for the number 27 pertains to finding whole numbers that multiply together to result in 27. Visual representation of factors can be conveyed via the multiplication process, best understood through the lens of pair factors. Below, a concise list of 27’s factors is provided:

• 1 × 27 = 27
• 3 × 9 = 27

Thus, the factors of 27 are 1, 3, 9, and 27.

To find the factors of 27 through division, follow these steps:

1. Divide 27 by numbers starting from 1 up to 27.
2. If the division results in a whole number without any remainder, then the divisor is a factor.

Here is a simplified representation of the process:

• 27 ÷ 1 = 27 (Factor is 1)
• 27 ÷ 3 = 9 (Factor is 3)
• 27 ÷ 9 = 3 (Factor is 9)
• 27 ÷ 27 = 1 (Factor is 27)

Any other division results in a non-whole number, hence they are not factors of 27.

Pair factors of 27 are two numbers that, when multiplied together, equal 27. Here is a table listing both positive and negative pair factors of 27:

The prime factorization of 27 involves breaking it down to its prime factors. The process is visualized through a factor tree which branches out until only prime numbers are left. For 27, the factor tree would repeatedly divide by the prime number 3:

• 27 ÷ 3 = 9
• 9 ÷ 3 = 3
• 3 ÷ 3 = 1

Thus, the prime factorization of 27 is 3 × 3 × 3, also written as 3³.

• The total count of 27’s factors is four: 1, 3, 9, and 27.
• The prime factor of 27 is 3.
• The pair factors of 27 are (1, 27) and (3, 9), along with their negative counterparts.
• The sum of all factors of 27 (1 + 3 + 9 + 27) is 40.

This exploration reveals that 27 is a perfect cube, as it can be expressed as the cube of its prime factor, 3.