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Factors of 105: As Taught by a Math Tutor
Methods
What are the Factors of 105?
As your math tutor, I’m here to guide you through the different types of factors of 105. Understanding factors is key to mastering multiplication, division, and prime numbers. Here’s a breakdown:
• 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 main ways a math tutor would explain how to find the factors of 105: using factor pairs and prime factorization. Let’s explore both!
The Factor Pairs of 105
As your math tutor, I’m here to help you break down factor pairs of 105 step by step!
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
Now you’ve got it! A math tutor would always encourage you to practice with different numbers to reinforce your understanding of factor pairs. Try another one!
Prime Factorization of 105
To find the prime factorization of 105, we break it down step by step until only prime factors remain. Then, we express 105 as a product of these prime factors multiplied together. Let’s go through the process and simplify it like a math tutor would!
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
Math Tutor Suggests: Find the Factors of Other Numbers
Practice your factoring skills by exploring how to factor other numbers, like the ones below:
Factors of 118 - The factors of 118 are 1, 2, 59, 118
Factors of 28 - The factors of 28 are 1, 2, 4, 7, 14, 28
Factors of 144 - The factors of 144 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
Factors of 16 - The factors of 16 are 1, 2, 4, 8, 16
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