Math Tutor Explains: What are the Factors of 840?

As your math tutor, I’m here to guide you through the different types of factors of 840. Understanding factors is key to mastering multiplication, division, and prime numbers. Here’s a breakdown:

• Factors of 840: 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840

• Sum of Factors of 840: 2880

• Negative Factors of 840: -1, -2, -3, -4, -5, -6, -7, -8, -10, -12, -14, -15, -20, -21, -24, -28, -30, -35, -40, -42, -56, -60, -70, -84, -105, -120, -140, -168, -210, -280, -420, -840

• Prime Factors of 840: 2, 3, 5, 7

• Prime Factorization of 840: 2^3 × 3^1 × 5^1 × 7^1

There are two main ways a math tutor would explain how to find the factors of 840: 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 840 step by step!

Factor pairs of 840 are any two numbers that, when multiplied together, equal 840. The question to ask is “what two numbers multiplied together equal 840?” Every factor can be paired with another factor, and multiplying the two will result in 840.

To find the factor pairs of 840, follow these steps:

Step 1:

Find the smallest prime number that is larger than 1, and is a factor of 840. 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 2.

Step 2:

Divide 840 by the smallest prime factor, in this case, 2:

840 ÷ 2 = 420

2 and 420 will make a new factor pair.

Step 3:

Repeat Steps 1 and 2, using 420 as the new focus. Find the smallest prime factor that isn’t 1, and divide 420 by that number. In this case, 2 is the new smallest prime factor:

420 ÷ 2 = 210

Remember that this new factor pair is only for the factors of 420, not 840. So, to finish the factor pair for 840, you’d multiply 2 and 2 before pairing with 210:

2 x 2 = 4

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 840:

(1, 840), (2, 420), (3, 280), (4, 210), (5, 168), (6, 140), (7, 120), (8, 105), (10, 84), (12, 70), (14, 60), (15, 56), (20, 42), (21, 40), (24, 35), (28, 30)

So, to list all the factors of 840: 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840

The negative factors of 840 would be: -1, -2, -3, -4, -5, -6, -7, -8, -10, -12, -14, -15, -20, -21, -24, -28, -30, -35, -40, -42, -56, -60, -70, -84, -105, -120, -140, -168, -210, -280, -420, -840

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 840, we break it down step by step until only prime factors remain. Then, we express 840 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 840 only has a few differences from the above method of finding the factors of 840. 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 840:

Step 1:

Find the smallest prime number that is larger than 1, and is a factor of 840. 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 2.

Step 2:

Divide 840 by the smallest prime factor, in this case, 2

840 ÷ 2 = 420

2 becomes the first number in our prime factorization.

Step 3:

Repeat Steps 1 and 2, using 420 as the new focus. Find the smallest prime factor that isn’t 1, and divide 420 by that number. The smallest prime factor you pick for 420 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 840 are: 2, 3, 5, 7

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

Factors of 69 - The factors of 69 are 1, 3, 23, 69

Factors of 139 - The factors of 139 are 1, 139

Factors of 24 - The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24

Factors of 106 - The factors of 106 are 1, 2, 53, 106