What are all the factors, the prime factorization, and factor pairs of 63?

Factors are integers we can multiply together to produce another number. They are essential for understanding the composition and properties of numbers in mathematics. Taking 63 as an example, we explore its factors which are the integers that, when multiplied in pairs, give us the product of 63. This scrutiny offers insight into the structure and attributes of 63 within the realm of numbers.

The concept of factors for a particular number like 63 involves identifying all the whole numbers that divide 63 without leaving a remainder. To visually represent these factors, we can list them in pairs demonstrating the multiplication process that results in 63. Here is a list of factors for 63:

• 1
• 3
• 7
• 9
• 21
• 63

Locating the factors of 63 through division is straightforward. We begin by dividing 63 by integers that could be potential factors since factors are equal to or smaller than the number itself. Suppose a division’s quotient is a whole number; then the divisor is indeed a factor of 63. To illustrate:

• 63 ÷ 1 = 63
• 63 ÷ 3 = 21
• 63 ÷ 7 = 9
• 63 ÷ 9 = 7
• 63 ÷ 21 = 3
• 63 ÷ 63 = 1

When 63 is divided by any number other than these, it results in a fraction. Thus, the factors of 63 are 1, 3, 7, 9, 21, and 63.

Pair factors of 63 are two numbers that when multiplied with one another yield the product of 63. Below are both the positive and negative pair factors for 63.

Positive Pair Factors	Negative Pair Factors
(1, 63)	(-1, -63)
(3, 21)	(-3, -21)
(7, 9)	(-7, -9)

The process of breaking down 63 into its prime factors is known as prime factorization. Here is a step-by-step demonstration along with a factor tree diagram representation:

63 can be divided by its smallest prime factor which is 3. Repeatedly dividing by 3, we get:

• 63 ÷ 3 = 21
• 21 ÷ 3 = 7

Since 7 is also a prime factor and cannot be divided further, the prime factorization of 63 is 3 x 3 x 7 or 3² x 7.

In summary, there are several key points to remember about the factors of 63:

• There are six positive factors of 63: 1, 3, 7, 9, 21, and 63.
• The prime factors of 63 are 3 and 7.
• The sum of all factors of 63 is 1 + 3 + 7 + 9 + 21 + 63 = 104.
• 63 is a composite number and hence has more than just two factors.