Math Tutor Explains: What are the Factors of 1728?

As your math tutor, I’m here to guide you through the different types of factors of 1728. Understanding factors is key to mastering multiplication, division, and prime numbers. Here’s a breakdown:

• Factors of 1728: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864, 1728

• Sum of Factors of 1728: 5080

• Negative Factors of 1728: -1, -2, -3, -4, -6, -8, -9, -12, -16, -18, -24, -27, -32, -36, -48, -54, -64, -72, -96, -108, -144, -192, -216, -288, -432, -576, -864, -1728

• Prime Factors of 1728: 2, 3

• Prime Factorization of 1728: 2^6 × 3^3

There are two main ways a math tutor would explain how to find the factors of 1728: 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 1728 step by step!

Factor pairs of 1728 are any two numbers that, when multiplied together, equal 1728. The question to ask is “what two numbers multiplied together equal 1728?” Every factor can be paired with another factor, and multiplying the two will result in 1728.

To find the factor pairs of 1728, follow these steps:

Step 1:

Find the smallest prime number that is larger than 1, and is a factor of 1728. 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 1728 by the smallest prime factor, in this case, 2:

1728 ÷ 2 = 864

2 and 864 will make a new factor pair.

Step 3:

Repeat Steps 1 and 2, using 864 as the new focus. Find the smallest prime factor that isn’t 1, and divide 864 by that number. In this case, 2 is the new smallest prime factor:

864 ÷ 2 = 432

Remember that this new factor pair is only for the factors of 864, not 1728. So, to finish the factor pair for 1728, you’d multiply 2 and 2 before pairing with 432:

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 1728:

(1, 1728), (2, 864), (3, 576), (4, 432), (6, 288), (8, 216), (9, 192), (12, 144), (16, 108), (18, 96), (24, 72), (27, 64), (32, 54), (36, 48)

So, to list all the factors of 1728: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864, 1728

The negative factors of 1728 would be: -1, -2, -3, -4, -6, -8, -9, -12, -16, -18, -24, -27, -32, -36, -48, -54, -64, -72, -96, -108, -144, -192, -216, -288, -432, -576, -864, -1728

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 1728, we break it down step by step until only prime factors remain. Then, we express 1728 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 1728 only has a few differences from the above method of finding the factors of 1728. 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 1728:

Step 1:

Find the smallest prime number that is larger than 1, and is a factor of 1728. 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 1728 by the smallest prime factor, in this case, 2

1728 ÷ 2 = 864

2 becomes the first number in our prime factorization.

Step 3:

Repeat Steps 1 and 2, using 864 as the new focus. Find the smallest prime factor that isn’t 1, and divide 864 by that number. The smallest prime factor you pick for 864 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 1728 are: 2, 3

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

Factors of 27 - The factors of 27 are 1, 3, 9, 27

Factors of 96 - The factors of 96 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96

Factors of 49 - The factors of 49 are 1, 7, 49

Factors of 64 - The factors of 64 are 1, 2, 4, 8, 16, 32, 64