 # Factors of 121: Prime Factorization, Methods, Tree, and Examples

The factors of 121 are factors that are entirely divisible by 121. 121 is an odd composite. It is also a perfect square of 11. Figure 1 – All possible Factors of 121

In this article, we will look at the different methods for finding out the factors of 121; factors of any number divide the number so that no remainder is produced upon division.

## What Are the Factors of 121?

There are only three factors of 121, which are 1, 11, and 121. These numbers yield a zero remainder when 121 is divided from them. The number 121 has only three factors because the number 121 acts as the perfect square of 11. So 2-factor pairs are formed by the factors of 121.

## How To Calculate the Factors of 121?

The factors of 121 can be found through the division method, prime factorization method, and drawing a factor tree. The prime factorization method is detailed in the next section, so that we will look at the division method first. The division method depends entirely on the complete division of the given number; if no remainder is produced upon the division of 121 with a certain number, then this certain number is a factor of 121. Moreover, the quotient, if it is a whole number, is also a factor after division. The entire method is shown below in detail for a better understanding of this concept:

Step 1: $\frac{121}{ 2} = 60.5$

60.5 is not a whole number, therefore 2 is not a factor of 121.

Step 2: $\frac{121}{ 3 }= 40.33$

Again, we can see that 40.33 is not a whole number, so 3 is also not a factor of 121.

Step 3: $\frac{121}{ 4 }= 30.25$

4 is also not a factor of 121 because 30.25 is not a whole number.

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Step 4: $\frac{121 }{ 11} = 11$

At this step, it is visible that 121 is a perfect square.

Step 5: $\frac{121}{ 121} = 1$

All numbers are factors of 1 and the number itself. From this method, we have found the following factors: Factors of 121 = 1, 11, and 121.

## Factors of 121 by Prime Factorization

Prime factorization is just another method for calculating prime factors. This method is based on simple division, but the major difference is that the quotient produced in one step shall act as the dividend for the next step of prime factorization.  Another difference is that we are looking for the result equal to zero, which is also our cue to stop the factorization process. All the divisors must always be prime numbers; this makes them prime factors of any given number. Prime factors are numbers 1, 2, 3, 5, 7, 11, and so on.  A certain number’s range of prime factors will always be between 1 and the half of 121. The factor will never be more significant than half of the number 121.  Prime factorization is an iterative process. Because we keep changing the dividend and divisor, the function between them stays the same, which is of simple division. This can be comprehended in more detail through the prime factorization of 121:

121 $\div$ 11 = 11

11 is the only number 121 is divisible, other than 1 and 121.

11 $\div$ 11 = 1

We notice that quotient 11 will now act as the dividend for this step; we have obtained one as the final answer, which means that our prime factorization has ended. This procedure can be explained through a basic equation below:

Prime Factorization of 121 = 11 x 11

The figure below also shows the prime factorization of 121: Figure 2 – Prime Factorization of 121

## Factor Tree of 121

The factor tree can be used as a visual aid for finding the prime factorization of 121. It is an extremely simple method which has been explained for 121 as follows: Step 1: Write down 121 and draw two branches from it; divide 121 by its prime factor in rough work, and the answer will be 11 when divided by 11. Step 2: On the ends of the two branches coming out from 121, write 11 on each back. Step 3: 11 can not be further divided by any other number as it itself is a prime number; therefore, this is where the factor tree ends. Learning from the steps explained above, look at this diagram to get a better understanding of the factor tree of 121. The Factor tree for 121 is given below: Figure 3 – Factor Tree of 121

## Factors of 121 in Pairs

Factor pairs are two factors of any random number which multiply to give the original random number. This is also the condition for any two numbers to be a factor pair. These factor pairs can be positive and negative. Let us look at the positive factor pairs of 121: Positive factor pairs of 121 =  (1, 121), (11, 11) The negative factor pairs of 121 are: Negative factor pairs of 121 = (-1, -121), (-11, -11) What is evident from those positive and negative factor pairs shown above is that they are the exact numbers. Still, the difference comes from the positive or negative sign attached to it.  We ignore the positive (+) sign in positive factor pairs, but in negative factor pairs, we need to attach the minus (-) sign with every number.

## Factors of 121 Solved Examples

We can not understand the factorization of 121 through theory only, so in this section, we will look at some examples that will allow us further to improve our understanding of the factorization of 121.

### Example 1

Note down the factors of 121 through the division and prime factorization methods, then compare the two.

### Solution

We will first find out the factors of 121 through the division method. As studied earlier, we divide the given number from any number to locate a factor in this method.  Now 121 is a perfect square of 11, so we know that 11 must be a factor of 121. In this example, we will use the trial and error method to check whether it has any other factors. For a divisor to be a factor, the division must produce no remainder, as we will see below: We start by dividing 121 by 2:

Step 1: $\dfrac{121}{ 2} = 60.5$

Two can not be a factor as it produces a decimal quotient.

Step 2: $\dfrac{121}{ 3} = 40.33$

Three can also not be a factor of 121 as it again produces a decimal quotient.

Step 3: $\dfrac{121}{ 4} = 30.25$

30.25 is a decimal number; 4 is also not a quotient.

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Step 4: $\dfrac{121}{ 11} = 11$

After 4, we jump to 11 because there are no factors after four, either. Here, we see that a whole number quotient is produced. Therefore, 11 is a factor of 121.

Step 5: $\dfrac{121}{ 121} = 1$

All numbers are completely divisible by themselves, so 121 is a factor along with 1. Through the above steps, we found out that 121 has only one factor other than 1 and 121 itself, and this factor is 11. Therefore the characteristics of 121 according to the division method are as follows: Factors of 121 through division method = 1, 11, and 121. Now moving on to the prime factorization method, we have the method as follows:

121 $\div$ 11 = 11

11 is the only number 121 is divisible, other than 1 and 121.

11 $\div$ 11 = 1

We notice that quotient 11 will now act as the dividend for this step; we have obtained one as the final answer, which means that our prime factorization has come to an end) So the prime factors of 121 are:

Prime factorization of 121 = 11 x 11

Differences between the two: The most notable difference between both methods is that we use all numbers to determine the factors in the division method. Still, in the prime factorization method, we only use prime numbers to divide the number.  Moreover, in factorization, we also use the quotient as a dividend in the next step, so that is a difference.

### Example 2

Find out the common factors in 121 and 66.

### Solution

The factors of 121 are given below: Factors of 121 = 1, 11, and 121. Similarly, the factors of 66 through the division method come out to be as follows: Factors of 66 = 1, 2, 3, 6, 11, 22, 33, and 66. From the factors of 121 and 66 mentioned above, we can see that only 1 and 11 are the common factors between 121 and 66. Therefore the common factors are: Common factors of 121 and 66 = 1, and 11. All images/mathematical drawings are created with GeoGebra.