What are the Factors of 105?

The following are the different types of factors of 105:

• Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105

• Sum of Factors of 105: 192

• Negative Factors of 105: -1, -3, -5, -7, -15, -21, -35, -105

• Prime Factors of 105: 3, 5, 7

• Prime Factorization of 105: 3^1 × 5^1 × 7^1

There are two ways to find the factors of 105: using factor pairs, and using prime factorization.

Factor pairs of 105 are any two numbers that, when multiplied together, equal 105. The question to ask is “what two numbers multiplied together equal 105?” Every factor can be paired with another factor, and multiplying the two will result in 105.

To find the factor pairs of 105, follow these steps:

Step 1:

Find the smallest prime number that is larger than 1, and is a factor of 105. For reference, the first prime numbers to check are 2, 3, 5, 7, 11, and 13. In this case, the smallest factor that’s a prime number larger than 1 is 3.

Step 2:

Divide 105 by the smallest prime factor, in this case, 3:

105 ÷ 3 = 35

3 and 35 will make a new factor pair.

Step 3:

Repeat Steps 1 and 2, using 35 as the new focus. Find the smallest prime factor that isn’t 1, and divide 35 by that number. In this case, 5 is the new smallest prime factor:

35 ÷ 5 = 7

Remember that this new factor pair is only for the factors of 35, not 105. So, to finish the factor pair for 105, you’d multiply 3 and 5 before pairing with 7:

3 x 5 = 15

Step 4:

Repeat this process until there are no longer any prime factors larger than one to divide by. At the end, you should have the full list of factor pairs.

Here are all the factor pairs for 105:

(1, 105), (3, 35), (5, 21), (7, 15)

So, to list all the factors of 105: 1, 3, 5, 7, 15, 21, 35, 105

The negative factors of 105 would be: -1, -3, -5, -7, -15, -21, -35, -105

To find the Prime factorization of 105, we break down all the factors of 105 until we are left with only prime factors. We then express n as a product of multiplying the prime factors together.

The process of finding the prime factorization of 105 only has a few differences from the above method of finding the factors of 105. Instead of ensuring we find the right factor pairs, we continue to factor each step until we are left with only the list of smallest prime factors greater than 1.

Here are the steps for finding the prime factorization of 105:

Step 1:

Find the smallest prime number that is larger than 1, and is a factor of 105. For reference, the first prime numbers to check are 2, 3, 5, 7, 11, and 13. In this case, the smallest factor that’s a prime number larger than 1 is 3.

Step 2:

Divide 105 by the smallest prime factor, in this case, 3

105 ÷ 3 = 35

3 becomes the first number in our prime factorization.

Step 3:

Repeat Steps 1 and 2, using 35 as the new focus. Find the smallest prime factor that isn’t 1, and divide 35 by that number. The smallest prime factor you pick for 35 will then be the next prime factor. If you keep repeating this process, there will be a point where there will be no more prime factors left, which leaves you with the prime factors for prime factorization.

So, the unique prime factors of 105 are: 3, 5, 7

Practice your factoring skills by exploring how to factor other numbers, like the ones below:

Factors of 108 - The factors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108

Factors of 77 - The factors of 77 are 1, 7, 11, 77

Factors of 112 - The factors of 112 are 1, 2, 4, 7, 8, 14, 16, 28, 56, 112

Factors of 33 - The factors of 33 are 1, 3, 11, 33