Factors of 693

The numbers that divide 693 evenly are known as factors of 693.

A factor of 693 is a number that divides the number without leaving a remainder.

The factors of 693 are 1, 3, 7, 9, 21, 33, 77, 99, 231, and 693.

Negative factors of 693: -1, -3, -7, -9, -21, -33, -77, -99, -231, and -693.

Prime factors of 693: 3, 7, and 11.

Prime factorization of 693: 3 × 3 × 7 × 11.

The sum of factors of 693: 1 + 3 + 7 + 9 + 21 + 33 + 77 + 99 + 231 + 693 = 1174

Factors can be found using different methods. Mentioned below are some commonly used methods:

• Finding factors using multiplication
   
• Finding factors using the division method
   
• Prime factors and prime factorization

To find factors using multiplication, we need to identify the pairs of numbers that are multiplied to give 693. Identifying the numbers which are multiplied to get the number 693 is the multiplication method.

Step 1: Multiply 693 by 1, 693 × 1 = 693.

Step 2: Check for other numbers that give 693 after multiplying:

3 × 231 = 693

7 × 99 = 693

9 × 77 = 693

21 × 33 = 693

Therefore, the positive factor pairs of 693 are: (1, 693), (3, 231), (7, 99), (9, 77), (21, 33).

For every positive factor, there is a negative factor.

Divide the given number with whole numbers until the remainder becomes zero and list out the numbers that result as whole numbers as factors. Factors can be calculated by following a simple division method:

Step 1: Divide 693 by 1, 693 ÷ 1 = 693.

Step 2: Continue dividing 693 by the numbers until the remainder becomes 0.

693 ÷ 1 = 693

693 ÷ 3 = 231

693 ÷ 7 = 99

693 ÷ 9 = 77

693 ÷ 21 = 33

Therefore, the factors of 693 are: 1, 3, 7, 9, 21, 33, 77, 99, 231, 693.

The factors can be found by dividing it with prime numbers. We can find the prime factors using the following methods:

• Using prime factorization
   
• Using a factor tree

Using Prime Factorization: In this process, prime factors of 693 divide the number to break it down in the multiplication form of prime factors till the remainder becomes 1.

693 ÷ 3 = 231

231 ÷ 3 = 77

77 ÷ 7 = 11

11 ÷ 11 = 1

The prime factors of 693 are 3, 7, and 11.

The prime factorization of 693 is: 3 × 3 × 7 × 11.

The factor tree is the graphical representation of breaking down any number into prime factors. The following step shows:

Step 1: Firstly, 693 is divided by 3 to get 231.

Step 2: Now divide 231 by 3 to get 77.

Step 3: Then divide 77 by 7 to get 11. Here, 11 is the smallest prime number, that cannot be divided anymore.

So, the prime factorization of 693 is: 3 × 3 × 7 × 11.

Factor Pairs Two numbers that are multiplied to give a specific number are called factor pairs.

Both positive and negative factors constitute factor pairs.

Positive factor pairs of 693: (1, 693), (3, 231), (7, 99), (9, 77), and (21, 33).

Negative factor pairs of 693: (-1, -693), (-3, -231), (-7, -99), (-9, -77), and (-21, -33).

• Factors: The numbers that divide the given number without leaving a remainder are called factors. For example, the factors of 693 are 1, 3, 7, 9, 21, 33, 77, 99, 231, and 693.
   
• Prime factors: The factors which are prime numbers. For example, 3, 7, and 11 are prime factors of 693.
   
• Factor pairs: Two numbers in a pair that are multiplied to give the original number are called factor pairs. For example, the factor pairs of 693 are (1, 693), (3, 231), etc.
   
• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 693 is 3 × 3 × 7 × 11.
   
• Multiplication method: A method to find factors by identifying pairs of numbers that multiply to a given number. For example, 3 and 231 are a multiplication pair for 693.