Factors of 85: Prime Factorization, Methods, Tree, and Examples

Factors of 85 are the numbers that, when multiplied in pairs, result in 85. Using multiple methods for calculating the factors of 85, we will look at how 85 can be divided into prime factors and normal factors, which give 85 when multiplied together.
factors of eighty five

Figure 1 – All possible Factors of 85

We will also show solved examples in the last section to make the lesson clear and comprehensive.

What Are the Factors of 85?

The factors of 85 are the following: 1, 5, 17, and 85. 85 has only four factors, so that they can be divided into two positive factor pairs and two negative factors. The number 85 is an odd composite number, so the factors it has are both prime and composite numbers. 

How To Calculate the Factors of 85?

The factors of 85 are easy to calculate through tools such as the division method. The division method is essentially based on finding factors of 85 through complete and even division. Numbers that divide 85 are entirely considered factors of it. Another method is the prime factorization of 85, which only gives the prime factors of 85; this method is visually represented and drawn through a factor tree diagram. Before getting into any other method, we will first look at the division method. As we discussed earlier, the division method is a straightforward approach to finding out the factors of 85. After identifying whether a number is even or odd, we start from 2 by dividing 85 by it to find whether it divides the number completely. We continue this process until we reach the possible range identified for the factors. This approach is shown in the steps below: \[ \frac{85}{ 5}  = 17 \]  We notice that 17 is the quotient produced after dividing 85 by 5. Now 17 does not have any decimal number, therefore 17 is a whole number. If the quotient is a whole number, then the divisor we chose is a factor. Hence, 5 is a factor. We look for the next factor now: \[ \frac{85}{ 17} = 5 \] 5 is also a whole number, so building on the logic explained above, 17 is also a factor of 85. This gives us another factor for 85, which is 17. Next, we divide 85 by itself: \[ \frac{85}{ 85} = 1 \] Hence, 85 is a factor as well. \[ \frac{85}{ 1 }= 85 \] 1 is also a factor of 85. Using this method, we find out the following factors of 85: Factors of 85 through division method: 1, 5, 17, and 85.

Factors of 85 by Prime Factorization

The factors of 85 calculated through prime factorization are called prime factors. Prime factors are obtained by dividing 85 with prime numbers only. The prime numbers which divide 85 ultimately and evenly are called prime factors. As we can observe, the division operation is also used in prime factorization but is not the same division we perform in basic calculations of mathematical numbers. This division involves constant changing of the divisor and the dividend. How this is done is explained in detail in this section: We find out the first prime number, which will divide 85 completely and evenly; this prime number turns out to be 5

85 $\div$ 5 = 17

Usually, we start from 2, which is also a prime number, but 85 is an odd number, so we know that 85 would not be divisible by 2. Moving on, the next prime number after 5 is 7, but 17, as we know, is not divisible by 7. The reason behind 17 not being divisible by 7 is because 17 itself is a prime number and is only divided completely by 1 and 17 itself, so:

17 $\div$ 17 = 1

Hence, both 5 and 17 are the prime factors of 85. We stopped here because dividing 17 by 17 gave us 1, and no further division is possible at this stage. Therefore, the prime factorization equation of 85 is:

Prime factorization of 85 = 5 x 17

Using the figure attached below, you can visually understand how we went about this method:
Prime factorization of eighty five

Figure 2 – Prime Factorization of 85

Factor Tree of 85

A factor tree is a schematic drawing in the shape of a tree. This tree is a visual representation of prime factorization where a tree-based diagram is used to symbolize the process of prime factorization. In prime factorization, we find out the prime factors of any number, which in this specific case is 85. So extrapolating from that, a factor tree also ends up at the prime factors of the given number. In the case of 85, the factor tree will end at only the prime factors of it. The diagram is started by writing down 85, and now we know that 85 is divisible by 5, which produces the number 17. So two branches extend out of 85. At the end of one, we write 5 and 17. On the other, both are prime numbers. Our factor tree ends here.  In figure 2, you can see for yourself how a factor tree can be constructed for any given number; that number, however, is 85 in our case:
Factor tree of eighty five

Figure 3 – Factor Tree of 85

Factors of 85 in Pairs

All factor pairs of 85 are essentially a pair of its factors written down between round brackets “()” and separated by a comma. Not any couple of random factors can make a factor pair of 85; there exists a necessary condition that has to be fulfilled. The condition is that when these two factors are multiplied by each other, they must produce an answer that is equal to the original number of which the factors were calculated. First, recall the factors of 85: Factors of 85 are the following: 1, 5, 17, and 85. To find out the factor pairs:

1 x 85 = 85

5 x 17 = 85

These are the positive factor pairs of 85: Factor pairs of 85 = (1, 85), (5, 17) Without any significant changes in the numbers, the negative factor pairs are positive factor pairs but with an inverted sign which means that positive (+) is changed to negative (-): Negative factor pairs of 85 = (-1, -85), (-5, -17)

Factors of 85 Solved Examples

Here are some examples that have been solved to provide an idea of how to go about factoring numbers.

Example 1

Write down the factors of 85 and 40. Also, find out the highest and smallest common factor between them.

Solution

The factors 85 and 40 can be found through the division method, and these factors are: The factors of 85 are: Factors of 85 are the following: 1, 5, 17, and 85. And the factors of 40 are: Factors of 40 = 1, 2, 4, 5, 8, 10, 20, and 40. We have the factors of both 85 and 40. To find out the highest common factor, let us first write down the common factors between the two. These common factors are 1 and 5.  Among 1 and 5, we can quickly point out that 5 is the highest common factor between 85 and 40. Similarly, 1 is the smallest common factor between 85 and 40.

Example 2

Use the prime factorization method to find out the prime factors of 80 and 85, then separate the factors of both into prime and composite numbers.

Solution

In this question, we are explicitly asked to calculate the prime factors of 85 and 80. The prime factorization has been studied in quite some detail in the previous section, so we will now just dive into solving it for 85 and 80. Solving for prime factors of 85:

85 $\div$ 5 = 17

17 $\div$ 17 = 1

Prime factors of 85 are 5 and 17. Solving for factors of 80:

80 $\div$ 2 = 40

40 $\div$ 2 = 20

20 $\div$ 2 = 10

10 $\div$ 2 = 5

5 $\div$ 5 = 1

These are the prime factors of 80: 2 x 2 x 2 x 2 x 5 In the next part of the question, we are asked to separate the prime and composite factors of 85 and 80. As you can recall, prime factorization produces prime factors; therefore, there are no composite numbers in the factor list of 85 and 80. Therefore: 85 and 80 have no composite factors through prime factorization, whereas the prime factors are 2, 5, and 17.

Example 3

What are the factors of 70? Also, calculate the sum of its factors.

Solution

We will first find out the factors of 70 through the division method; these factors turn out to be: Factors of 70 = 1, 2, 5, 7, 10, 14, 35, and 70. In the next step, we will calculate the sum of all its factors:

Sum of factors of 70 = 1 + 2 + 5 + 7 + 10 + 14 + 35 + 70

Sum of factors of 70 = 144

Therefore, the sum of factors of 70 is equal to 144. All images/mathematical drawings are created with GeoGebra. 

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