Factors of -135

The numbers that divide -135 evenly are known as factors of -135. A factor of -135 is a number that divides the number without a remainder.

The factors of -135 are 1, 3, 5, 9, 15, 27, 45, and 135.

Negative factors of -135: -1, -3, -5, -9, -15, -27, -45, and -135.

Prime factors of -135: 3 and 5.

Prime factorization of -135: 3³ × 5¹ 

The sum of positive factors of 135 (ignoring the negative sign): 1 + 3 + 5 + 9 + 15 + 27 + 45 + 135 = 240

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 -135. Identifying the numbers which are multiplied to get the number -135 is the multiplication method.

Step 1: Multiply -135 by 1, -135 × 1 = -135.

Step 2: Check for other numbers that give -135 after multiplying: 3 × -45 = -135 5 × -27 = -135 9 × -15 = -135 (Additionally consider negative factor pairs similarly)

Therefore, the positive factor pairs of -135 are: (1, 135), (3, 45), (5, 27), (9, 15). For every positive factor, there is a corresponding negative factor.

Dividing the given numbers with whole numbers until the remainder becomes zero and listing out the numbers which result as whole numbers as factors. Factors can be calculated by following the simple division method:

Step 1: Divide -135 by 1, -135 ÷ 1 = -135.

Step 2: Continue dividing -135 by the numbers until the remainder becomes 0. -135 ÷ 1 = -135 -135 ÷ 3 = -45 -135 ÷ 5 = -27 -135 ÷ 9 = -15

Therefore, the factors of -135 are: 1, 3, 5, 9, 15, 27, 45, 135.

The factors can be found by dividing them 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 -135 divide the number to break it down in the multiplication form of prime factors till the remainder becomes 1. -135 ÷ 3 = -45 -45 ÷ 3 = -15 -15 ÷ 3 = -5 -5 ÷ 5 = -1 The prime factors of -135 are 3 and 5.

The prime factorization of -135 is: 3³ × 5¹

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

Step 1: Firstly, -135 is divided by 3 to get -45.

Step 2: Now divide -45 by 3 to get -15.

Step 3: Then divide -15 by 3 to get -5.

Step 4: Divide -5 by 5 to get -1.

Here, -1 is the smallest factor, and 3 and 5 are prime numbers, that cannot be divided anymore.

So, the prime factorization of -135 is: 3³ × 5¹.

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 -135: (1, 135), (3, 45), (5, 27), (9, 15).

Negative factor pairs of -135: (-1, -135), (-3, -45), (-5, -27), (-9, -15).

• Factors: The numbers that divide the given number without leaving a remainder are called factors. For example, the factors of -135 are 1, 3, 5, 9, 15, 27, 45, 135, and their negatives.

• Prime factors: The factors which are prime numbers. For example, 3 and 5 are prime factors of -135.

• 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 -135 are (1, -135), (3, -45), etc.

• Negative factors: Negative numbers that divide the given number without leaving a remainder. For example, -1, -3, -5 are negative factors of -135.

• Prime factorization: Expressing a number as a product of its prime factors. For example, the prime factorization of -135 is \(3^3 \times 5 \times -1\).