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Factors of 77: Prime Factorization, Methods, Tree, and Examples

Factorization of 77 is the division of 77 into factors that multiply to give the original number again. All factors of 77 are the numbers that perform a complete division of 77, meaning they do not leave any remainder upon division. 

The methods for finding out these factors are mentioned in this article in detail. 

What Are the Factors of 77?

There are only four factors of the number 77, which are 1, 7, 11, and 77. These four factors indicate that 77 is a composite number.

All these numbers generate zero as the remainder when they act as divisors for 77. These factors can be positive and negative. The four factors can be split into 2-factor pairs.

How To Calculate the Factors of 77?

The factors of 77 can be calculated through the division method, prime factorization method, and the factor tree method, which is also prime factorization but represented through a tree.

These methods are easy to follow and will be within our complete grasp by the end of this article. From these methods, let us first discuss the division method, which is the most traditional and straightforward of all. 

Before starting the division method, it is essential to find out the range of factors acquired through division, so we have an inkling of how many numbers we are supposed to check for factorization.

The range of factors can be calculated by dividing the original number 77 by 2.

\[ \frac{77} {2} \approx 39 \]

So we know from this division that the factors of 77 will always be within 1 to 39 other than the factor 77 itself. We will now look at the division method in detail for number 77.

In this step, we will divide 77 by the closest possible divisor, which in this case is 7; upon dividing 77 by 7, we have 11, which is a whole number. We know that 7 is a complete divisor and does not produce any remainder; therefore, seven is one of our factors for 77

\[ \frac{77}{ 7} = 11\]

In this step, we look for the next possible divisor after 7; after going through the number list, this number comes out to be 11, so we will divide 77 by 11; upon division, we see that we get a quotient equal to 7, which is also a whole number. Therefore 11 divides 77 completely and is a factor of 77:

\[ \frac{77}{ 11} = 7 \]

We now look for any other divisor within the range of 1 to 39, but no other number divides 77 completely; therefore, no other factor exists for 77 other than 7 and 11. Finally, for clarity’s sake, we will divide 77 by 77 and show that 77 is, in fact, a factor of 77 itself:

\[ \frac{77}{ 77} = 1 \]

Using this method of division, we have calculated the following factors of 77:

Factors of 77 through the division method = 1, 7, 11, and 77.

Factors of 77 by Prime Factorization

Prime factorization is also a division-based method for calculating the factors of 77. In prime factorization, we find out the factors of 77 through division by prime numbers. The numbers which divide 77 are entirely called prime factors.

This is not a simple division like the above; the difference is that in prime factorization, once we perform a step of dividing 77 by any prime number, the resulting quotient from this division will now act as the dividend. 

Unlike the division method, where we keep dividing 77 by different numbers. So the new quotient will act as a dividend for step 2. This can be understood more from the procedure given below:

Step 1: For prime factorization, in step 1, we will divide 77 by the smallest possible prime number that divides it completely; in our case, this prime number comes out to be 7. To recall, prime numbers are the numbers only divisible by themselves, and 1:

77 $\div$ 7 = 11

Step 2: The 11 from step 1 will now act as the dividend for this step, which means that we will now look for a prime number divisor that can divide 11 completely without any remainder. This number is 11 itself because 11 again is a prime number:

11 $\div$ 11 = 1

Step 3: In the previous step, we obtained the final result equal to 1 in any prime factorization of any number; if we achieve 1, our method has ended, and we can not divide any further. 

This is the last step of prime factorization, where we note the prime factors obtained through this method. These prime factors are:

Prime factors of 77 = 7, 11 

If we use an equation to represent this method, it would be:

Prime Factorization of 77 = 7 x 11

To look at the prime factorization of 77, look below at the figure attached:

Figure 1

Factor Tree of 77

The factor tree of 77 is a visual diagram representing the prime factors of 77. This diagram is built in a shape similar to that of a tree but only restricted to two branches per step; in each step, we perform prime factorization and divide the branch into two further branches. 

This diagram for the factor tree of prime factorization of 77 can be seen below in figure 2:

Figure 2

Factors of 77 in Pairs

Factor pairs of 77 are such two factors that multiply to give back 77. These pairs are unique in that not every factor, coupled with another factor, will give 77. 

It is a property special to only some of the factors of 77. This can be seen clearly by looking at the factors of 77:

Factors of 77 = 1, 7, 11, and 77.

Let us multiply 7 with 1 to find out if 7 and 1 make a factor pair, but their multiplication answers 7, which is not equal to 77. Therefore 7 and 1 do not make a factor pair. Working on this condition, the factor pairs of 77 come out to be the following:

Factor pairs of 77 = (1, 77), (7, 11)

As 77 has only four factors, it will have only two-factor pairs; all numbers have positive and negative factor pairs. Positive factor pairs of 77 are:

Positive factor pairs of 77 = (1, 77), (7, 11)

Similarly, the negative factor pairs of 77 are:

Negative factor pairs of 77 = (1, 77), (7, 11)

Factors of 77 Solved Example

Using this section, you will learn how to apply the theory of factorization of 77 through multiple methods on real-life mathematical problems. This will strengthen your learning of the topic, so it is encouraged that you go through these at least once.

Example 1

Using the methods of prime factorization and division, write down the factors of 77 and its prime factors.

Solution

We will first perform the division method to find out the factors of 77; it must be recalled that in the division method, we keep dividing the number 77 with numbers within its factors range to find out the entire list of factors, so we proceed as follows:

Step 1: 77 $\div$ 7 = 11 

As we know that 11 is a whole number; therefore, 7 divides it completely, making seven a factor of 77.

Step 2: 77 $\div$ 11 = 7 

7 is also a whole number; therefore, we can deduce easily that 11 is also a factor of 77 as it gives zero remainders upon division with 77.

Step 3: 77 $\div$ 77 = 1 

As we can see in this step, all numbers are factors.

Step 4: 77 $\div$ 1 = 77 

All numbers are also factors of 1, which is a universal factor.

Now we have calculated the factors of 77 through the division method; these factors are:

Factors of 77 through division method = 1, 7, 11, and 77.

The next task is to find out the prime factors of 77, so we proceed as follows:

Step 1: We will divide 77 by any prime number that divides it completely; in our case, this prime number is 7:

77 $\div$ 7 = 11

Step 2: The 11 from step 1 will now act as the dividend for step 2, which means that we will now look for the next smallest prime number divisor that can divide 11 completely without producing any remainder. This number is 11 itself because 11 again is a prime number:

11 $\div$ 11 = 1

So the prime factors of 77 are:

Prime factorization of 77 = 7 x 11

All images/mathematical drawings are created with GeoGebra. 

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