What are all the factors, the prime factorization, and factor pairs of 144?

Factors are the fundamental building blocks in mathematics that constitute a number. They are the integers which, when multiplied together, produce the number in question. Understanding the factors of a number, such as 144, helps us explore the mathematical relationships and properties inherent in that number. Factors reveal the divisibility and multiplicativity of numbers, useful in various branches of mathematics including algebra and number theory.

The concept of factors pertains to numbers which, when multiplied together in pairs, yield the number 144. To visualize this concept, consider the multiplication process where two integers combine to form 144. Below is a list of factors that, upon combination, result in the number 144:

• 1 x 144 = 144
• 2 x 72 = 144
• 3 x 48 = 144
• 4 x 36 = 144
• 6 x 24 = 144
• 8 x 18 = 144
• 9 x 16 = 144
• 12 x 12 = 144

To find the factors of 144, one can employ the division method. Here, we divide 144 by integers less than or equal to the number until we find pairs that divide without a remainder. The divisor in each pair is a factor of 144. The process is as such:

144 ÷ 1 = 144 (1 is a factor)
144 ÷ 2 = 72 (2 is a factor)
144 ÷ 3 = 48 (3 is a factor)
…
144 ÷ 144 = 1 (144 is a factor)

When 144 is divided by any other number not listed, the result is a fraction, indicating that the number is not a factor.

The full set of factors of 144, determined through division, are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, and 144.

Pair factors of 144 consist of two numbers that, when multiplied together, give the product of 144. Below are the positive and negative pair factors:

The prime factorization of 144 involves breaking the number down into its basic prime factors. The process can be visualized using a factor tree diagram, which simplifies 144 into its prime components. The factor tree for 144 is as follows:

144 → 2 × 72 → 2 × (2 × 36) → 2 × (2 × (2 × 18)) → 2 × (2 × (2 × (2 × 9))) → 2 × 2 × 2 × 2 × 3 × 3

Therefore, the prime factorization of 144 is 2^4 × 3^2.

• Total count of factors of 144: 15
• Prime factors of 144: 2 and 3
• Pair factors of 144: (1, 144), (2, 72), (3, 48), (4, 36), (6, 24), (8, 18), (9, 16), (12, 12)
• The sum of all factors of 144: 403

Note that the number 144 is a perfect square since it can be expressed as 12^2, which also implies that (12, 12) is both a pair factor and a reiteration of 144’s square root.

1. List all the factors of 144 that are also multiples of 12.
2. What is the product of the prime factors of 144?
3. Which pair factors of 144 represent it as a square of an integer?

1. The factors of 144 that are also multiples of 12 are: 12, 24, 36, 48, 72, and 144.
2. The product of the prime factors of 144, considering their multiplicity, is 2^4 × 3^2, which equals 144.
3. The pair factors of 144 that represent it as a square of an integer are: (12, 12), since 12^2 = 144.