What are all the factors, the prime factorization, and factor pairs of 84?

Factors are the building blocks of numbers; they are the integers we multiply together to produce another number. Understanding these factors illuminates the structure and characteristics of a number within the realm of mathematics. Let’s consider 84 as an illustrative example to delve into the world of factors. Factors of 84 are the whole numbers which, when multiplied in pairs, give the product as 84. This exploration offers insights into the makeup and properties of 84 in the mathematical landscape.

The concept of factors for the number 84 revolves around identifying those integers that can evenly divide 84 without leaving a remainder. Below is a list of factors for 84 that showcases this concept clearly.

• 1
• 2
• 3
• 4
• 6
• 7
• 12
• 14
• 21
• 28
• 42
• 84

Discovering the factors of 84 can be achieved through the division method. This approach involves dividing 84 by integers that are less than or equal to it. The division method is depicted as follows:

• Divide 84 by 1: 84 ÷ 1 = 84
• Divide 84 by 2: 84 ÷ 2 = 42
• Divide 84 by 3: 84 ÷ 3 = 28
• Divide 84 by 4: 84 ÷ 4 = 21

If the result of a division is a whole number, that divisor is a factor of 84.

Pair factors of 84 are two numbers which, when multiplied together, equal 84. These can be positive or negative and are listed below:

Negative pair factors are also applicable since the multiplication of two negative numbers yields a positive product. For example, -1 × -84 = 84.

The prime factorization of 84 involves breaking it down into its prime factors. This can be visually represented through a factor tree, which simplifies 84 step-by-step into its prime elements.

Thus, the prime factorization of 84 is 2 × 2 × 3 × 7, which can also be written as 2² × 3 × 7.

Here are some vital points regarding the factors of 84 to keep in mind:

• Total factors: 12
• Prime factors: 2, 3, and 7
• Pair factors: Both positive and negative pair factors such as (1, 84) and (-1, -84)
• Sum of all factors: The sum of all factors of 84 is 224.

An interesting fact about 84 is that it is both a product of a twin prime (3 and 7) and a perfect square (2²).