Find all the factors of:
729

Answer

To find all the factors of 729, we can follow these steps:

### Step 1: Prime Factorization
First, we need to perform the prime factorization of 729. We start by dividing 729 by the smallest prime number, which is 3.

- 729 ÷ 3 = 243
- 243 ÷ 3 = 81
- 81 ÷ 3 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

So, we can express 729 as:
\[ 729 = 3^6 \]

### Step 2: Finding Factors
Now that we have the prime factorization, we can find all the factors of 729. The factors can be obtained by taking all combinations of the prime factors raised to their respective powers.

The powers of 3 that can be used are from 0 to 6:
- \( 3^0 = 1 \)
- \( 3^1 = 3 \)
- \( 3^2 = 9 \)
- \( 3^3 = 27 \)
- \( 3^4 = 81 \)
- \( 3^5 = 243 \)
- \( 3^6 = 729 \)

### Step 3: Listing All Factors
Now we can list all the factors of 729:
- 1
- 3
- 9
- 27
- 81
- 243
- 729

Thus, the complete list of factors of 729 is:
\[ \{1, 3, 9, 27, 81, 243, 729\} \]

### Summary
The factors of 729 are:
1, 3, 9, 27, 81, 243, and 729.

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