Math Tutor Explains: What are the Factors of 315?

As your math tutor, I’m here to guide you through the different types of factors of 315. Understanding factors is key to mastering multiplication, division, and prime numbers. Here’s a breakdown:

• Factors of 315: 1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105, 315

• Sum of Factors of 315: 624

• Negative Factors of 315: -1, -3, -5, -7, -9, -15, -21, -35, -45, -63, -105, -315

• Prime Factors of 315: 3, 5, 7

• Prime Factorization of 315: 3^2 × 5^1 × 7^1

There are two main ways a math tutor would explain how to find the factors of 315: using factor pairs and prime factorization. Let’s explore both!

As your math tutor, I’m here to help you break down factor pairs of 315 step by step!

Factor pairs of 315 are any two numbers that, when multiplied together, equal 315. The question to ask is “what two numbers multiplied together equal 315?” Every factor can be paired with another factor, and multiplying the two will result in 315.

To find the factor pairs of 315, follow these steps:

Step 1:

Find the smallest prime number that is larger than 1, and is a factor of 315. 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 315 by the smallest prime factor, in this case, 3:

315 ÷ 3 = 105

3 and 105 will make a new factor pair.

Step 3:

Repeat Steps 1 and 2, using 105 as the new focus. Find the smallest prime factor that isn’t 1, and divide 105 by that number. In this case, 3 is the new smallest prime factor:

105 ÷ 3 = 35

Remember that this new factor pair is only for the factors of 105, not 315. So, to finish the factor pair for 315, you’d multiply 3 and 3 before pairing with 35:

3 x 3 = 9

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 315:

(1, 315), (3, 105), (5, 63), (7, 45), (9, 35), (15, 21)

So, to list all the factors of 315: 1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105, 315

The negative factors of 315 would be: -1, -3, -5, -7, -9, -15, -21, -35, -45, -63, -105, -315

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!

To find the prime factorization of 315, we break it down step by step until only prime factors remain. Then, we express 315 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 315 only has a few differences from the above method of finding the factors of 315. 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 315:

Step 1:

Find the smallest prime number that is larger than 1, and is a factor of 315. 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 315 by the smallest prime factor, in this case, 3

315 ÷ 3 = 105

3 becomes the first number in our prime factorization.

Step 3:

Repeat Steps 1 and 2, using 105 as the new focus. Find the smallest prime factor that isn’t 1, and divide 105 by that number. The smallest prime factor you pick for 105 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 315 are: 3, 5, 7

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

Factors of 148 - The factors of 148 are 1, 2, 4, 37, 74, 148

Factors of 63 - The factors of 63 are 1, 3, 7, 9, 21, 63

Factors of 88 - The factors of 88 are 1, 2, 4, 8, 11, 22, 44, 88

Factors of 144 - The factors of 144 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144