What are all the factors, the prime factorization, and factor pairs of 54?

Factors are the fundamental numbers that we can multiply together to obtain a specific product, in this case, 54. This concept is a cornerstone in the world of mathematics as it allows us to explore the compositional aspects and intrinsic properties of numbers such as 54. Understanding the factors of a number can aid in various mathematical operations and problem-solving scenarios.

Factors of 54 are the whole numbers that evenly divide 54 without leaving any remainder. Visually representing these factors can illuminate the relationships between them, as seen through their multiplication pairings. Below, we provide a comprehensive enumeration of the factors of 54:

• 1 x 54
• 2 x 27
• 3 x 18
• 6 x 9
• 9 x 6
• 18 x 3
• 27 x 2
• 54 x 1

To discover the factors of 54, we can employ the division method. This practical approach involves dividing 54 by an integer to see if the quotient is a whole number, which indicates that the integer is a factor. Any number that produces a fraction cannot be a factor. Here is how the division method unfolds for 54:

• 54 ÷ 1 = 54
• 54 ÷ 2 = 27
• 54 ÷ 3 = 18
• 54 ÷ 6 = 9
• 54 ÷ 9 = 6
• 54 ÷ 18 = 3
• 54 ÷ 27 = 2
• 54 ÷ 54 = 1

As a result, the factors of 54 we have identified are 1, 2, 3, 6, 9, 18, 27, and 54.

Pair factors of 54 are the pairs of positive or negative numbers that, when multiplied together, equate to 54. It is interesting to observe that like all numbers, 54 has both positive and negative pair factors. The table below lists the pair factors of 54:

Prime factorization breaks down the number 54 into its prime factors. A prime factor is a prime number that multiplies with other primes to give the original number. Let’s deconstruct 54 into prime factors using a factor tree:

54 ÷ 2 = 27

27 ÷ 3 = 9

9 ÷ 3 = 3

Then, we are left with all prime numbers: 2, 3, and 3. The prime factorization of 54 is 2 x 3 x 3 x 3 or 2 x 3³.

When reviewing the factors of 54, it is essential to note the following points:

• The sum of all factors of 54 is 120.
• 54 has a total of 8 factors: 1, 2, 3, 6, 9, 18, 27, and 54.
• The prime factors of 54 are 2 and 3.
• The pair factors of 54 come in positive and negative pairs: (1, 54), (2, 27), (3, 18), (6, 9), and their respective negative pairs.

These points help us gain a comprehensive understanding of the number 54 and its factor-related properties.

Try to solve the following questions related to the factors of 54:

• Q1. What is the second smallest factor of 54?
• Q2. List all the factors of 54 that are also prime numbers.
• Q3. What is the product of the prime factors of 54?

Below are the solutions to the exercises on the factors of 54:

• A1. The second smallest factor of 54 is 2.
• A2. The factors of 54 that are prime numbers are 2 and 3.
• A3. The product of the prime factors of 54, considering their multiplicity, is 54 itself (2 x 3 x 3 x 3 = 54).