Factors of 98: Prime Factorization, Method, Tree, And Examples

The factors of 98 are the numbers that give 98 as the product when these numbers are multiplied together. Additionally, the factors of 98 also include the numbers which give zero as the remainder whenever 98 is divided from such numbers.
Factors of ninety eight

Figure 1 – All possible Factors of 98

The number 98 is an even composite number which is an indication of the fact that the number 98 consists of more than 2 factors. Since numerous possibilities exist for the factors of 98, an easy way to determine the factors of 98 is to determine the set of numbers where the greatest number is the number itself. The set containing the numbers begins from the smallest factor 1 and ends at half of that number. In the case of the number 98, the half of 98 is 49 so this indicates that the range of factors of 98 will lie between 1 and 49. All the numbers lying between 1 and 49 can be a factor of 98, keeping in mind the condition that the number must leave zero as the remainder and a whole number quotient when it acts as the divisor. Multiple techniques can be used to determine these factors such as the division method and the prime factorization method. This article is all about a detailed analysis of various methods that are used to determine the factors of number 98.

What Are the Factors of 98?

The factors 98 are 1, 2, 7, 14, 49, and 98. These are the numbers that yield zero as the remainder when they act as the divisors for the number 98.  The number 98 has a total of 6 factors and these factors can be both positive and negative. The smallest factor in the list is 1 and the largest is 98. 

How To Calculate the Factors of 98?

You can calculate the factors of 98 by looking for the numbers that leave zero as the remainder when 98 is divided from such numbers. This seems like a tedious process but it can be made easier with the help of some simple methods. The most used and most accurate method for determining the factors of any number is the division method. The division method states that if a number leaves zero as the remainder and yields a whole number quotient when it acts as the divisor, then that number can be labeled as a factor. For determining the factors of 98, we already know the range of these factors, which is from 1 to 49, so let’s imply the division method to all possible factors. Since 98 is an even number so let’s take a look at the division of 98 with 2: \[ \frac{98}{2} = 49 \] As a whole number quotient is produced, hence 2 acts as the factor of 98. Another special feature of the division method is that the whole number quotient produced also acts as a factor. This is evident from the division shown below: \[ \frac{98}{49} = 2\] Again, a whole number quotient is produced so both 2 and 49 are the factors of 98. Additional factors of 98 are given below: \[ \frac{98}{1} = 98\] \[ \frac{98}{7} = 14 \] \[ \frac{98}{14} = 7 \] \[ \frac{98}{98} = 1\] Hence, all the factors of 98 are given below: Factors of 98: 1, 2, 7, 14, 49, 98 These factors can be negative as well. The negative factors are the same as positive factors, the only difference is the negative sign. The negative factors of 98 are given below: Negative Factors of 98: -1, -2, -7, -14, -49, -98

Factors of 98 by Prime Factorization

Prime factorization is a simple division technique that is used to determine the prime factors for any given number. The prime factors are the numbers that act as the factors for a given number and are also prime numbers. The prime factorization begins with the number itself. This number undergoes division with a prime number and yields a whole number quotient. This quotient then acts as the dividend and goes through the division with a prime number. This process is repeated until 1 is achieved at the end. The presence of 1 in the division indicates the termination of the prime factorization. The prime factorization of the number 98 is shown below:

98 $\div$ 2 = 49

49 $\div$ 7 = 7 

7 $\div$ 7 = 1

So the prime factorization of the number 98 is:

Prime Factorization of 98 = 2 x 7 x 7

OR

Prime Factorization of 98 = 2 x $7^{2}$

The prime factors obtained as the result of the prime factorization are given below: Prime Factors = 2, 7 The prime factorization of 98 is also shown in figure 1 given below: 
Prime factorization of ninety eight

Figure 2 – Prime Factorization of 98

Factor Tree of 98

The factor tree is a visual representation of the prime factorization of the number. Just like the prime factorization, the factor tree is also used to determine the prime factors for a number. But instead of ending at 1, as in the case of prime factorization, the factor tree ends at prime numbers. The factor tree begins with the number and then splits into 2 branches – one branch showing the prime factor and the other branch showing the whole number quotient produced as the result of the division.  The factor tree for the number 98 is shown below:
Factor tree of ninety eight

Figure 3 – Factor Tree of 98

Factors of 98 in Pairs

The factors of 98 can also exist in the form of factor pairs. These pairs are the two numbers that when multiplied together give the original number as the product. Since the number 98 consists of 6 factors, so these 6 factors can be split into 3-factor pairs. The numbers within the pair must give 98 as the product when they are multiplied together. The factor pairs of 98 are given below:

1 x 98 = 98

2 x 49 = 98

7 x 14 = 98

Factor Pairs = (1, 98), (2, 49), and (7, 14) These factor pairs can be negative as well. The negative factor pairs are similar to the positive ones, the only difference is that the negative factor pairs have a negative sign with them.  The condition for negative factor pairs is that both numbers must have negative signs so when they are multiplied, a positive product is obtained. The negative factors of 98 are given below:

-1 x -98 = 98

-2 x -49 = 98

-7 x -14 = 98

Negative Factor Pairs = (-1, -98), (-2, -49), and (-7, -14)

Factors of 98 Solved Examples

Some solved examples are given below regarding the factors of 98 to help strengthen your understanding.

Example 1

Calculate the average of both the even and the odd factors of 98.

Solution

For calculating the average, let’s first list these factors of 98. The factors of 98 are given below: Factors of 98: 1, 2, 7, 14, 49, 98 Now, let’s first calculate the average of the even factors. The even factors of 98 are given below: Even factors of 98 = 2, 14, 98 Calculating their sum:

Sum of even factors of 98 = 2 + 14 + 98

Sum of even factors of 98 =  114

The average of the even factors of 98 is given below: \[ Average = \frac{Sum}{\text{Total Number of factors}} \] \[ Average = \frac{114}{3} \]

Average = 38

Hence the average of the even factors of 98 is 39. Now, let’s calculate the average of the odd factors. The odd factors of 98 are given below: Odd factors of 98 = 1, 7, 49 Calculating their sum:

Sum of odd factors of 98 = 1 + 7 + 49

Sum of odd factors of 98 = 57

The average of odd factors of 98 is given below: \[ Average = \frac{Sum}{\text{Total Number of factors}} \]

Average = $\dfrac{57}{3}$ 

Average = 19

So the average of odd factors of 98 is 19.

Example 2

Calculate the sum of all the factors of 98. Determine if this sum is divisible by 2.

Solution

For calculating the sum of factors of 98, let’s first list them down. Factors of 98 = 1, 2, 7, 14, 49, 98 Calculating the sum of these factors:

Sum = 1 + 2 + 7 + 14 + 49 + 98

Sum = 171

Now, since the resulting sum is an odd number, hence this number is not divisible by 2.

Example 3

Calculate the product of all the even factors of 98.

Solution

To calculate the product of all the even factors of 98, let’s first list down all the factors of 98. Factors of 98 = 1, 2, 7, 14, 49, 98 Now, let’s take a look at the even factors of 98. The even factors of 98 are given below: Even Factors of 98 = 2, 14, 98 Calculating their product:

Product of even factors of 98 = 2 x 14 x 98

Product of even factors of 98 = 2744

So the product of the even factors of 98 is 2744. As this number is an even number, this number is a multiple of 2.  All images/mathematical drawings are created with GeoGebra. 

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