Factors of 70: Prime Factorization, Methods, Tree, And Examples

The factors of 70 are the group of factors that give zero as the remainder and a whole number quotient whenever 70 is divided from such numbers. The divisor and their respective whole number quotients act as the factors for such divisors.
Factors of seventy

Figure 1 – All possible Factors of 70

Factors of 70 are also the numbers that produce 70 when these two numbers are multiplied together. These two numbers together constitute a factor pair, which is a pair of factors for any number, in this case, 70. The number 70 is even composite. Since it is a composite number, that naturally means that the number 70 has more than 2 factors. A fun fact about the number 70 is that its last digit is zero, indicating that 70 is a multiple of both 5 and 10. The division of 70 with ten is shown below: \[ \frac{70}{10} = 7\] A whole number quotient is produced as a result of this division which indicates that both 10 and 7 are factors of 70.  The factors of 70 are both positive as well as negative. This article is all about conducting a detailed evaluation of the factors of 70 and the various methods which are used to determine these factors. So let’s dive right in!

What Are the Factors of 70?

The factors of 70 are 1, 2, 5, 7, 10, 14, 35, and 70. These factors can be both positive as well as negative. All these numbers yield zero as the remainder when they act as the divisor for the number 70. The number 70 has a total of 8 factors, with 70 being the largest factor. 

How To Calculate the Factors of 70?

You can calculate the factors of 70 through various techniques, such as the division method, multiplication, and prime factorization method. Let’s have a brief introduction to each technique. The division method is one of the most common and easiest ways to determine the factors. The condition for the division method is the divisor should produce zero as the remainder and a respective whole number quotient. In such a case, the divisor and the whole number quotient act as the factors for that number. For the number 70, let’s first consider the division of 70 with the smallest factor, 1. This division is shown below: \[ \frac{70}{1} = 70 \] Since a whole number quotient is produced, this indicates that both 1 and 70 are factors of 70. To validate this statement, the division of 70 with the whole number quotient produced is shown below: \[\frac{70}{70} = 1\] The division of additional factors of 70 is shown below: \[ \frac{70}{2} = 35 \] \[ \frac{70}{5} = 14 \] \[ \frac{70}{7} = 10 \] \[ \frac{70}{10} = 7 \] \[ \frac{70}{14} = 5 \] \[ \frac{70}{35} = 2 \] So, the list of all the factors of 70 is given below:

Factors List of 70: 1, 2, 5, 7, 10, 14, 35, and 70

These factors can be negative as well. The negative factors of 70 are given below:

Negative Factors of 70: -1, -2, -5, -7, -10, -14, -35, and -70

Factors of 70 by Prime Factorization

Prime factorization is a methodology used to determine the factors of any given number. Prime factorization is mainly used to find prime factors of the number. Prime factors are the prime numbers that also act as the factors for a given number. The methodology of prime factorization is similar to the division method, but the only difference is that in prime factorization, only prime numbers carry out the division process. This process of division continues until 1 is achieved at the end.  The prime factorization begins with the number itself, which undergoes division with a prime number. The resulting whole number quotient acts as the dividend, and the process repeats. This division with prime number goes all the way till at the end 1 is obtained. The process of prime factorization of 70 is shown below:

70 $\div$ 7 = 10

10 $\div$ 5 = 2

2 $\div$ 2 = 1 

So the prime factorization of 70 can be written as follows:

Prime Factorization of 70 = 2 x 5 x 7

The prime factorization of 70 is also shown in figure 2 given below:
Prime factorization of seventy

Figure 2 – Prime Factorization of 70

Hence, according to the prime factorization of 70, the following prime factors are obtained for 70:

Prime Factors = 2, 5, 7

Factor Tree of 70

A factor tree is a pictorial way to represent the prime factors for a given number. In a factor tree, the number is presented in the form of its prime factors and the respective whole number quotient produced.  The division of numbers in a factor tree is carried out similarly to prime factorization. The only difference is that instead of ending at 1, as in the prime factorization, the factor tree ends at prime numbers. The factor tree begins with the number itself and then extends its branches into a prime factor and a respective whole number quotient. This whole number quotient acts as the dividend and spreads its branches into a prime factor and a whole number quotient. This process continues till prime numbers are obtained at the end branches. The factor tree for the number 70 is shown below in figure 3:
Factor tree of seventy

Figure 3 – Factor Tree of 70

Factors of 70 in Pairs

A factor pair of 70 can be defined as a pair of numbers that not only act as the factors but also produce the original number when multiplied. A factor pair consists of 2 numbers. In the case of 70, there are 8 factors in existence which indicates that the number 70 will have a total of 4-factor pairs. These factor pairs are formed between the divisor and the whole number quotient. For instance, let’s consider the division of 70 with 7: \[ \frac{70}{7} = 10 \] Since ten is produced as the whole number quotient, 7 forms a factor pair with 10. This is because when these two numbers are multiplied, 70 is produced as the result.

7 x 10 = 70

Additional factor pairs for the number 70 are given below:

2 x 35 = 70

5 x 14 = 70

1 x 70 = 70

The factor pairs of 70 are given below:

Factor Pairs of 70: (1, 70), (2, 35), (5, 14), and (7, 10)

These factor pairs can be negative as well. The only condition for negative factor pairs is that both the numbers existing within the pair must have a negative sign to yield a positive number when multiplied. The negative factor pairs of 70 are given below:

-1 x -70 = 70

-2 x -35 = 70

-5 x -14 = 70

-7 x -10 = 70

So the negative factor pairs of 70 are given below:

Negative Factor Pairs = (-1, -70), (-2, -35), (-5, -14), and (-7, -10)

Factors of 70 Solved Examples

To further strengthen the concept of the factors of 70, given below are a few examples that can help to develop an understanding of the factors of 70.

Example 1

Find the sum of the odd and even factors of 70 and determine their average, respectively.

Solution

Let’s first list these factors to determine the sum of the even and odd factors of 70. The factors of 70 are given below:

Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70

Now, let’s determine the sum of even factors of 70. For this purpose, let’s list down the even factors of 70:

Even factors of 70: 2, 10, 14, 70

Their sum is given below:

Sum of even factors of 70 = 2 + 10 + 14 + 70

Sum of even factors of 70 = 96

So, the sum of even factors of 70 is 96. Now, let’s determine the sum of the odd factors of 70. These odd factors of 70 are given below:

Odd Factors of 70 = 1, 5, 7, 35

Their sum is given below:

Sum of odd factors of 70 = 1 + 5 + 7 + 35

Sum of odd factors of 70 = 48

Now that we have determined the sum of even and odd factors of 70, let’s determine their average.

Average of even factors = $\frac{\text{Sum of even factors}}{\text{Number of even factors}}$

Average of even factors = $\frac{96}{4}$

Average of even factors = 24

So the average of even factors of 70 is 24. Now, let’s determine the average of the odd factors of 70:

Average of odd factors = $\frac{\text{Sum of odd factors}}{\text{Number of odd factors}}$

Average of odd factors = $\frac{48}{4}$

Average of odd factors = 12

So the average of odd factors of 70 is 12.

Example 2

Calculate the sum of all the factors by 70.

Solution

For calculating the sum of all the factors of 70, let’s first list down these factors.

Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70

The sum of these factors is given below:

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

Sum = 144

Hence, the sum of all the factors of 70 is 144. 

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