[Maths Class Notes] on Factors of 81 Pdf for Exam

The Factors of 81 are all positive as well as the negative integers or whole numbers that you can evenly divide into the number 81 81. 81, when divided by a factor of 81, will equal another factor of the number 81. Factor pairs of 81 are the whole numbers that can be either positive or negative but not in the form of a fraction or a decimal number. 

The factors of 81 and the prime factors of 81 are different since 81 is a composite number. Also, even though they are closely related, the prime factors of the number 81 and the prime factorization of the number 81 are not exactly the same. In this lesson, we will take a look at what are the factors of 81, the prime factorization of 81, the factor pairs of 81, and the prime factor tree of 81.

The factors of 81 are the numbers that give the result as 81 when 2 numbers are multiplied with each other. For instance, the factor pair of 7 is written as (1,7) and (-1,-7).

When the pair of negative factors is multiplied, the result should give the original number, such as multiplying -1 × -7, then we get the number 7. Therefore, we can consider both positive and negative factor pairs of 7. Factor pairs of the number 81 are defined as the whole numbers that could be either positive or negative but not a fraction or decimal number. 

What are the Factors of 81?

Since all the factors of 81 are the numbers that you can evenly divide by the number 81, you simply need to divide this number 81 by all the numbers up to 81 and see which ones would result in an even quotient. When you do that, you will find that the following calculations resulted in the even quotient:

81 ÷ 1 = 81

81 ÷ 3 = 27

81 ÷ 9 = 9

81 ÷ 27 = 3

81 ÷ 81 = 1

The positive factors of the number 81 are therefore all the numbers that you used to divide above to and get an even number. Given below is the list of all the positive factors of 81 in the numerical order:

1, 3, 9, 27, 81.

The factors of 81 include all the negative numbers as well. Hence, all the positive factors of 81 can be easily converted to negative numbers. The list of the negative factors of the number 81 are:

-1, -3, -9, -27, -81.

Factors of 81 in Pairs

The factors of 81 in pairs are all the combinations of two different factors which when multiplied together would equal 81. Given below are all the positive factor pairs of 81.

1. 1 × 81 = 81

2. 3 × 27 = 81

3. 9 × 9 = 81

4. 27 × 3 = 81

5. 81 × 1 = 81

Since the factors of 81 include negative numbers as well, minus times minus equals plus, and thus you can easily convert the positive factor pairs of the number 81 by putting a minus sign in front of all the factors and getting all the negative factor pairs of 81.

1. -1 × -81 = 81

2. -3 × -27 = 81

3. -9 × -9 = 81

4. -27 × -3 = 81

5. -81 × -1 = 81

Prime Factor Tree of 81

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The prime factorization of 81 is given by 3 x 3 x 3 x 3. This is a different list of the prime factors and their multiplicities. Note that the prime factorization of the number 81 does not include the number 1, however, it does include every instance of the specific prime factor.

81 is the composite number. In contrast to the prime numbers that only have one factorization, composite numbers like 81 have a minimum of two factorizations.

To demonstrate what this means, select the rightmost and the leftmost integer in 81, 27, 9, 3, 1, and then multiply these integers to get the number 81. This is called the first factorization. 

Next, choose the second rightmost and the second leftmost number for obtaining the second factorization which also gives 81.

The prime factorization or the integer factorization of 81 means finding the set of the prime numbers that when multiplied together, would give you the original number 81. This is also referred to as the prime decomposition of 81.

Hence, the prime factors are written as 3 × 3 × 3 × 3 or 334 in which 3 is the prime number.

Calculate Factors of 81 by Prime Factorization Method

The factors of 81 can be calculated using the Prime Factorization method in 2 ways.

1. By the factor tree method

2. By the upside-down division method,

The prime factorization of 81 is 3 × 3 × 3 × 3 = 34. Therefore, the prime factor of 81 is 3, which can be multiplied to get the other factors.

Prime factorization of 81

The prime factors of a positive integer are defined as the prime numbers that divide the respective integer exactly. The prime factorization of a positive integer comprises a list of the integer’s prime factors along with their multiplicities. The process of determining these factors is called integer factorization.

A prime factorization is a result of factoring a number into a set of components in which every member is a prime number. This is generally written by showing 81 as a product of its prime factors. For 81, prime factorization would be:

81 = 3 x 3 x 3 x 3

Negative Factors of 81

The positive numbers can be flipped into negatives and those negative numbers would also be factors of 81. E.g. -1, -3, -9, -27, and -81

A factor pair is a combination of 2 factors that can be multiplied together to equal 81. All of the possible factor pairs of number 81 are given below:

Factors of 81 in Pairs

Number of factors of 81

The Division Method is used to find the number of factors.

For instance, 81 = 3 x 3 x 3 x 3 = 3 

4means 3 to the power 4

Number of factors is equal to, (power + 1)(power + 1)(power + 1)………. 

Hence, (4 + 1) = 5 

And if we consider both positive and negative factors, 

The total number of factors = 5 (positive) + 5 (negative) = 10

The highest common factor of 81

The greatest number that divides 81 and 42 exactly is their greatest common factor, 

For instance, GCF of 81 and 42.

Factors of 81 and 42:

Factors of 81 = 1, 3, 9, 27, 81.

Factors of 42 = 1, 2, 3, 6, 7, 14.

Highlights of Factors of 81

• The factors of 81 can be defined as the numbers which we multiply to get 81 

• Factors of 81 are 1, 3, 9, 27, and 81.

• (1, 81), (3, 27), and (9, 9) are the factors of 81 in pairs.

• 1 is a universal factor. It is a factor of all numbers. The number itself is called a factor of the number if it divides itself exactly.

Conclusion

This is all about the different types of factors of 81 and how to calculate them. There are ways to determine the different factors of 81 explained here. Focus on the explanation and grab the concepts well to learn how to determine factors of other numbers too.