The prime factorization of 729 is 3 multiplied by itself six times, which can be expressed as 3⁶.
Understanding Prime Factorization
Prime factorization is a fundamental concept in mathematics where a composite number is broken down into its prime factors. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7, 11). Every composite number has a unique set of prime factors.
How to Find the Prime Factorization of 729
To determine the prime factorization of 729, we can use the division method, repeatedly dividing the number by the smallest possible prime number until the quotient is 1.
Let's break down 729:
- Is 729 divisible by 2? No, because 729 is an odd number.
- Is 729 divisible by 3? Yes. The sum of its digits (7 + 2 + 9 = 18) is divisible by 3.
- 729 ÷ 3 = 243
- Is 243 divisible by 3? Yes. (2 + 4 + 3 = 9, which is divisible by 3).
- 243 ÷ 3 = 81
- Is 81 divisible by 3? Yes. (8 + 1 = 9, which is divisible by 3).
- 81 ÷ 3 = 27
- Is 27 divisible by 3? Yes.
- 27 ÷ 3 = 9
- Is 9 divisible by 3? Yes.
- 9 ÷ 3 = 3
- Is 3 divisible by 3? Yes.
- 3 ÷ 3 = 1
The process concludes when we reach 1. The prime factors are all the divisors used in this process.
Expressing the Prime Factors
From the steps above, the prime factors of 729 are 3, 3, 3, 3, 3, and 3.
Therefore, the prime factorization of 729 can be written as:
729 = 3 × 3 × 3 × 3 × 3 × 3
Using Exponents for Prime Factorization
For a more concise representation, especially when a prime factor appears multiple times, exponents are used. An exponent indicates how many times the base number is multiplied by itself. In the case of 729, the prime factor 3 is repeated 6 times.
Thus, 729 expressed in exponential form is:
729 = 3⁶
This notation clearly shows that 729 is the sixth power of 3.
Other Factors of 729
While 3 is the only prime factor of 729, the number itself has several other factors, which are combinations of its prime factors.
| Type of Factor | Examples |
|---|---|
| All Factors | 1, 3, 9, 27, 81, 243, 729 |
| Prime Factors | 3 |
| Pair Factors | (1, 729), (3, 243), (9, 81), (27, 27) |
The unique prime factor for 729 is indeed 3, and its prime factorization is 3 raised to the power of 6.
