What are all the factors, the prime factorization, and factor pairs of 9?

Factors are the building blocks of numbers, representing the whole numbers that can be multiplied together to produce the number in question. For our discussion, let’s consider the number 9 as a prime example. Factors of a number [Number], such as 9, reveal its divisibility and connection to other numbers in the field of mathematics.

The factors of [Number], or specifically 9 in this case, are integers that divide the number without leaving a remainder. In multiplication, certain pairs of these factors will result in the original number, 9. Below is a list showcasing the factor pairs of 9:

• 1 × 9 = 9
• 3 × 3 = 9

These pairings illustrate how the multiplication of these factors equals 9.

To identify the factors of 9, we utilize the division method. Division aids us in finding which numbers divide 9 evenly, making them its factors. Here are the steps to discover all the factors of 9:

• Begin by dividing 9 by the smallest integer 1, which will invariably result in the original number, thus 1 is a factor of everything.

• Continue with the division using increasing integers until you can no longer obtain a whole number as a result. For 9, the next integer is 3.

• 9 ÷ 3 = 3, which implies that both 3 and its quotient are factors of 9.

Any attempt to divide 9 by numbers other than these factors will result in a non-whole number. For instance, 9 ÷ 2 = 4.5. Since only whole numbers can be factors, 2 is not a factor of 9.

Thus, the full list of factors for 9 includes 1 and 9 itself, as well as 3. Hence, the factors of 9 are 1, 3, and 9.

Pair factors of 9 are sets of two numbers that, when multiplied together, equal 9. These pairs can encompass both positive and negative numbers, since the product of two negatives is a positive. Here are the pair factors of 9:

Positive or negative, these pairs exemplify the concept of factors and their role in defining the number 9.

Prime factorization involves breaking down a number into its basic prime factors. For 9, the process would unfold through multiple divisions by prime numbers until only 1 remains. Observe the diagram below for a visual depiction:

The prime factorization of 9 is simply 3 × 3 or 3², since 3 is the smallest prime number that divides 9 evenly.

Key takeaways from our exploration of the number 9’s factors include:

• The factors of 9 are 1, 3, and 9.
• Prime factors of 9: 3
• Pair factors of 9: (1, 9) and (3, 3)
• Prime factorization of 9: 3²
• The sum of 9’s factors: The sum of 1+3+9 equals 13.

These facts about 9 offer insight into its properties as a square number, being 3 squared, and its application in various mathematical scenarios.

• 1. List all the factors of 9.
• 2. Find the sum of the factors of 9.
• 3. How many pair factors does the number 9 have?

Here are the solutions to the previous exercises:

• 1. The factors of 9 are: 1, 3, and 9.
• 2. The sum of the factors of 9: 1 + 3 + 9 = 13.
• 3. The number 9 has two pair factors: (1, 9) and (3, 3).