What are all the factors, the prime factorization, and factor pairs of 48?

Factors are the numbers we multiply together to get a product. Understanding these factors sheds light on the structure and properties of a number within mathematics. Taking 48 as an illustrative example, we see that factors can reveal much about a number’s divisibility and its relationships with other numbers.

Factors of 48 are the set of whole numbers that can be multiplied together in pairs to produce 48. These factors include both the number itself and 1. The complete list of factors for 48 clearly demonstrates the concept and the various combinations that result in the given number.

• 1 x 48 = 48
• 2 x 24 = 48
• 3 x 16 = 48
• 4 x 12 = 48
• 6 x 8 = 48

We can easily determine the factors of 48 by employing the division method. This approach involves dividing 48 by various integers to determine which yield a whole number as the quotient. Factors of any number must be equal to or less than the original number; hence we explore divisions with 48 and smaller integers.

• 48 ÷ 1 = 48, so 1 and 48 are factors.
• 48 ÷ 2 = 24, so 2 and 24 are factors.
• 48 ÷ 3 = 16, so 3 and 16 are factors.
• 48 ÷ 4 = 12, so 4 and 12 are factors.
• 48 ÷ 6 = 8, so 6 and 8 are factors.
• And we continue in this manner until all possible factors are found.

Therefore, any other division apart from these will not result in whole numbers, indicating that no further factors exist.

Pair factors of 48 are the sets of two numbers that, when multiplied together, equal 48. Below is a table illustrating both the positive and negative pairing.

The prime factorization of 48 is breaking it down into its prime factors, which are the building blocks of the number. The process typically involves creating a factor tree. For 48, we proceed as follows:

• We start by dividing 48 by its smallest prime factor, which is 2.
• 48 ÷ 2 = 24, then 24 ÷ 2 = 12, followed by 12 ÷ 2 = 6, and finally, 6 ÷ 2 = 3.
• Since 3 is a prime number, we stop here.

Thus, the prime factorization of 48 is 2 x 2 x 2 x 2 x 3, or 2⁴ x 3.

In summary, 48 is a composite number that has notable factors and properties:

• Total factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. (10 factors in total)
• Prime factors of 48: 2 and 3.
• Pair factors of 48 include positive and negative numbers resulting in 48 when multiplied.
• The sum of all factors of 48 (excluding itself) is 93.

Interesting fact: 48 is a highly composite number due to its many divisors, which makes it a common multiple for many pairs of smaller numbers.

• Exercise 1: List the prime factors of 48.
• Exercise 2: Identify at least three pair factors of 48.
• Exercise 3: What is the sum of all factors of 48?

• Solution 1: The prime factors of 48 are 2 and 3.
• Solution 2: Pair factors of 48 include (1, 48), (2, 24), and (3, 16).
• Solution 3: The sum of all factors of 48 is 1+2+3+4+6+8+12+16+24+48 = 124.