 # Factors of 729

Factors of 729 are the integers that divide the original number (i.e. 729) completely. A factor divides the number into equal number of parts. The factors will be less than or equal to 729, but they could not be greater than the original number. Also, 729 is a perfect cube, such that;

729 = 9 x 9 x 9 = 93

By this, we can conclude that 9 is one of the factors of 729. To check, we can divide 729 by 9.

729 ÷ 9 = 81 [Remainder = 0]

Thus, 9 divides 729 into 81 equal parts.

Now let us learn to find the other factors along with pair factors and prime factors.

 Factors Pair Factors Prime Factorisation 1, 3, 9, 27, 81, 243 and 729 (1, 729), (3, 243), (9, 81), (27, 27) 729 = 3 x 3 x 3 x 3 x 3 x 3 = 36

## How to Find Factors of 729?

Factors of 729 are the numerical values that divide the original number, without leaving any remainder. Therefore, dividing 729 by the smallest natural number, 1, will result in the actual number. Since 729 is an odd number, it cannot be divided by 2 or any even numbers.

729 ÷ 1 = 729

729 ÷ 3 = 243

729 ÷ 9 = 81

729 ÷ 27 = 27

729 ÷ 81 = 9

729 ÷ 243 = 3

729 ÷ 729 = 1

Thus, we can conclude that only 1, 3, 9, 27, 81, 243 and 729 are the factors of 729.

## Pair Factors of 729

To find the pair factors of 729, we need to find the product of the two numbers such that they result in the original number.

1 × 729 = 729

3 × 243 = 729

9 × 81 = 729

27 × 27 = 729

Therefore, the factors in pairs for 729 are (1, 729), (3, 243), (9, 81), (27, 27).

Similarly, we can consider negative pair factors of 729, since the product of two negative numbers will result in a positive number.

-1 × -729 = 729

-3 × -243 = 729

-9 × -81 = 729

-27 × -27 = 729

Therefore, the negative pair factors are (-1, -729), (-3, -243), (-9, -81), (-27, -27).

## Prime Factorisation of 729

Prime factors are the prime numbers that can divide the original number, uniformly. Now to determine the prime factors of 729, we have to divide 729 by prime numbers, each time, until the quotient is 1.

Step 1: 729 is divisible by 3

729/3 = 243

Step 2: 243 is divisible by 3

243/3 = 81

Step 3: 81 is divisible by 3

81/3 = 27

Step 4: 27 is divisible by 3

27/3 = 9

Step 5: 9 is divisible by 3

9/3 = 3

Step 6: 3 is divisible by itself.

3/3 = 1

Step 7: Since 1 is not divisible by any other factor apart from itself. Hence,

 Prime factorisation of 729 = 3 x 3 x 3 x 3 x 3 x 3 = 36 Prime Factor is 3

## Solved Examples

Q.1: Rama bought 81 chocolates for Rs. 729. What is the cost of each chocolate?

Solution: Given,

Number of chocolates bought by Rama = 81

Cost of 81 chocolates = Rs. 729

Cost of 1 chocolate = 729/81 = 9.

Therefore, cost of each chocolate will be Rs. 9.

Q.2: Find the sum of all the factors of 729.

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

Sum = 1+3+9+27+81+243+729 = 1093

Therefore, 1093 is the required sum.

Q.3: What are the common factors of 99 and 729?

Answer: Let us list down the factors of both the given numbers.

The factors of 99 are: 1, 3, 9, 11, 33, 99

The factors of 729 are: 1, 3, 9, 27, 81, 243, 729

As we can see, the common factors of 99 and 729 are 1, 3 and 9.

## Frequently Asked Questions on Factors of 729

### How many factors of 729 are there?

There are seven factors of 729. They are 1, 3, 9, 27, 81, 243 and 729.

### What are the prime factors of 729?

The prime factor of 729 is 3, since the prime factorisation results in:

729 = 3 x 3 x 3 x 3 x 3 x 3

### What are the multiples of 729?

Multiples are the integers that are divisible by the original number. Thus, the first ten multiples of 729 are 729, 1458, 2187, 2916, 3645, 4374, 5103, 5832, 6561 and 7290.

### What is the LCM of 729 and 243?

LCM of 729 and 243 is 729.

By prime factorisation method we can write:

729 = 3 × 3 × 3 × 3 × 3 × 3

243 = 3 × 3 × 3 × 3 × 3

Thus, multiplying each factor the greatest number of times it occurred,

LCM (243, 729) = 3 × 3 × 3 × 3 × 3 × 3 = 729.

### Is 729 a perfect square?

729 is a perfect cube and not a perfect square, such that;

729 = 9 x 9 x 9 = 93

Thus, if we take the cube root of 729, the value will be equal to 9.