What are all the factors, the prime factorization, and factor pairs of 120?

Factors are the building blocks of numbers that, when multiplied together, yield the product of the original number. With 120 as our illustrative example, factors are the numbers we multiply to achieve 120. This exploration into the factors of a number informs us about its structure and relationships in the realm of mathematics, offering insight into its divisibility and the various combinations that constitute it.

Factors of a number, such as 120, are the integers that when multiplied in pairs give the product 120. Here is a list of factors for 120 depicted in both numerical and visual format for easier understanding.

• 1 × 120 = 120
• 2 × 60 = 120
• 3 × 40 = 120
• 4 × 30 = 120
• 5 × 24 = 120
• 6 × 20 = 120
• 8 × 15 = 120
• 10 × 12 = 120

The factors of any number, including 120, are found using the division method. We systematically divide 120 by positive integers to discover which divisions result in whole numbers without any remainder.

• 120 ÷ 1 = 120, so 1 is a factor of 120.
• 120 ÷ 2 = 60, thus 2 is a factor of 120.
• Continuing this pattern, we find that the numbers 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120 all divide 120 evenly.
• If we attempt to divide 120 by a number that is not a factor, such as 7, we end up with a fraction: 120 ÷ 7 = 17.14

Following this reliable method ensures that we find all factors of the number 120.

Pair factors of a number are two-factor pairs that when multiplied together give the number itself. For 120, both positive and negative pairs can be considered:

Note that the negative pair factors are also valid because the product of two negative numbers is positive.

The prime factorization of 120 involves breaking it down into its prime factors, which are the prime numbers that, when multiplied together, result in 120. The process can be visualized through a factor tree:

• Divide 120 by the smallest prime number possible, which is 2: 120 ÷ 2 = 60
• Continue to divide by 2 as long as the quotient is even: 60 ÷ 2 = 30, then 30 ÷ 2 = 15
• Proceed with the next smallest prime, which is 3: 15 ÷ 3 = 5
• As 5 is already a prime number, we conclude the factorization: 2 × 2 × 2 × 3 × 5 or 2³ × 3 × 5

Therefore, the prime factorization of 120 is 2³ × 3 × 5.

While exploring the factors of 120, keep the following points in mind:

• 120 has a total of 16 factors.
• The prime factors of 120 are 2 and 3.
• The sum of all the factors of 120 is 360.

Factors form an integral part of the number’s identity, providing insight into divisibility, multiplication, and more.

1. Find the common factors of 120 and 30.
2. Calculate the product of all the prime factors of 120.
3. Determine if 15 is a factor of 120 and why.

The solutions to the exercises are:

1. The common factors of 120 and 30 include 1, 2, 3, 5, 6, 10, 15, 30.
2. The product of all the prime factors of 120 is 2³ × 3 × 5 = 120.
3. Yes, 15 is a factor of 120, as 120 ÷ 15 = 8, which is a whole number without any remainder.