What are all the factors, the prime factorization, and factor pairs of 80?

Factors are the building blocks that multiply together to give a product. Understanding factors is crucial for delving into the properties and structure of numbers in mathematics. Let’s begin by exploring the factors of the number 80, which serve as a foundation for better grasping its arithmetical essence.

The factors of 80 are the whole numbers that multiply together to reach the product of 80. Pairs of these factors, when combined, equal the target number, serving as a clear representation of the concept. Below is a list of these factors for reference:

• 1
• 2
• 4
• 5
• 8
• 10
• 16
• 20
• 40
• 80

To discover the factors of 80, one can utilize the division method. This involves dividing 80 by potential factors, which are integers less than or equal to 80. If the division yields a whole number, the divisor is deemed a factor. Let’s inspect this process more closely:

1 and 80 are inherently factors of 80, as every number is divisible by 1 and itself. Since 80 is an even number, 2 is a factor—80 divided by 2 equates to 40. Continuing with this pattern:

• 80 ÷ 1 = 80
• 80 ÷ 2 = 40
• 80 ÷ 4 = 20
• 80 ÷ 5 = 16
• 80 ÷ 8 = 10
• 80 ÷ 10 = 8
• 80 ÷ 16 = 5
• 80 ÷ 20 = 4
• 80 ÷ 40 = 2
• 80 ÷ 80 = 1

Any other division of 80 by integers not listed above will result in a non-whole number—thus, not a factor.

Pair factors of 80 consist of two numbers which, when multiplied together, equal 80. This includes both positive and negative factor pairs. Here are the pair factors:

The prime factorization of 80 involves breaking down the number into its prime factors, which are the prime numbers that multiply to 80. To illustrate:

• 80 ÷ 2 = 40
• 40 ÷ 2 = 20
• 20 ÷ 2 = 10
• 10 ÷ 2 = 5

The prime factors of 80 are thus 2 and 5, and the prime factorization is 2^4 × 5 or 2 × 2 × 2 × 2 × 5.

When considering the factors of 80, it’s essential to note that 80 has a total of 10 factors, including 2 and 5 as its prime factors. The sum of all factors of 80 is 186. It’s interesting to observe that the factor pairs can be both positive and negative, which affirms the diverse relationships within multiplication involving 80.

Try these questions to reinforce your understanding of the factors of 80:

1. Identify a factor pair of 80 that sums up to 90.
2. What is the product of all the prime factors of 80?

Here are the solutions to the above exercises:

1. The factor pair of 80 that sums up to 90 is (10, 80), where 10 + 80 = 90.
2. The product of all the prime factors of 80 (considering multiplicity) is 2^4 × 5 = 80.