Factors of 84 | Prime Factorization of 84 | Factor Tree of 84

Factors of 84

Factors of 84	Factor Pairs of 84	Prime factors of 84
1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84	(1,84) (2,42) (3,28) (4,21) (6,14) (7,12) (12,7) (14,6) (21,4) (28,3) (42,2)	2 x 2 x 3 x7

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What are the factors of 84

The factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.

A factor is a number that divides evenly into another number. To find the factors of a number, we can start by dividing the number by 2 and then working our way up to the number itself, dividing by each whole number along the way to see if it is a factor. If a number is a factor, we can add it to our list of factors.

In the case of 84, we can do the following:

1. Divide 84 by 2: 84 / 2 = 42. Since 42 is a whole number, 2 is a factor of 84.
2. Divide 84 by 3: 84 / 3 = 28. Since 28 is a whole number, 3 is a factor of 84.
3. Divide 84 by 4: 84 / 4 = 21. Since 21 is a whole number, 4 is a factor of 84.
4. Continue dividing by the next whole number (in this case, 5) and checking the result. Repeat this process until you reach the number itself.

The complete list of factors of 84 is 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.

How to Find Factors of 84

Here are four methods that you can use to find the factors of 84:

1. Factors of 84 using the Multiplication Method
2. Factors of 84 using the Division Method
3. Prime Factorization of 84
4. Factor tree of 84

Factors of 84 Using the Multiplication Method

The factors of 84 can also be found using the multiplication method. To find the factors of a number using the multiplication method, we can pair up the numbers from 1 to the number itself and see which pairs multiply to equal the number.

For example, let’s say we want to find the factors of 84 using the multiplication method:

• Begin by pairing up the numbers from 1 to the number itself (in this case, 84).

The pairs are:

(1, 84)

(2, 42)

(3, 28)

(4, 21)

(6, 14)

(7, 12)

• Check which pairs multiply to equal the number.

The pairs that multiply to equal 84 are:

(1, 84)

(2, 42)

(3, 28)

(4, 21)

(6, 14)

(7, 12)

Therefore, the factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.

Factors of 84 Using the Division Method

To find the factors of a number using the division method, we can divide the number by each whole number from 2 to the number itself and see which ones result in a whole number.

For example, let’s say we want to find the factors of 84 using the division method:

• Begin by dividing the number by 2. If the result is a whole number, then 2 is a factor of the number.

For 84, 84 / 2 = 42, which is a whole number. Therefore, 2 is a factor of 84.

• Divide the number by the next whole number (in this case, 3). If the result is a whole number, then 3 is a factor of the number.

For 84, 84 / 3 = 28, which is a whole number. Therefore, 3 is a factor of 84.

• Divide the number by the next whole number (4 in this case). If the result is a whole number, then 4 is a factor of the number.

For 84, 84 / 4 = 21, which is a whole number. Therefore, 4 is a factor of 84.

• Continue dividing the number by the next whole number (5 in this case) and checking the result. Repeat this process until you reach the number itself.

The complete list of factors of 84 found using the division method is 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.

Prime Factorization of 84

The prime factorization of a number is the expression of that number as the product of its prime factors. To find the prime factorization of a number, we can divide the number by the smallest possible prime number (in this case, 2) and then continue dividing the result by the smallest possible prime number until we can’t divide anymore. The resulting prime factors can be multiplied together to get the original number.

For example, let’s find the prime factorization of 84:

• Begin by dividing the number by 2. If the result is a whole number, then 2 is a factor of the number.

For 84, 84 / 2 = 42, which is a whole number. Therefore, 2 is a factor of 84.

• Divide the result (42) by 2. If the result is a whole number, then 2 is a factor of the number.

For 42, 42 / 2 = 21, which is a whole number. Therefore, 2 is a factor of 42.

• Divide the result (21) by 2. If the result is a whole number, then 2 is a factor of the number.

For 21, 21 / 2 = 10.5, which is not a whole number. Therefore, 2 is not a factor of 21.

• Divide the result (21) by the next smallest prime number (in this case, 3). If the result is a whole number, then 3 is a factor of the number.

For 21, 21 / 3 = 7, which is a whole number. Therefore, 3 is a factor of 21.

• Divide the result (7) by the next smallest prime number (in this case, 5). If the result is a whole number, then 5 is a factor of the number.

For 7, 7 / 5 is not a whole number. Therefore, 5 is not a factor of 7.

• The prime factorization of 84 is 2 x 2 x 3 x 7.

Factor tree of 84

A factor tree is a tool that helps us find the prime factorization of a number. It allows us to break down a number into its prime factors by dividing it by the smallest possible prime number and then continuing to divide the result by the smallest possible prime number until we can’t divide anymore. The resulting prime factors can be multiplied together to get the original number.

To create a factor tree, we follow the same process as finding the prime factorization of a number:

• Begin by dividing the number by 2. If the result is a whole number, then 2 is a factor of the number.

For 84, 84 / 2 = 42, which is a whole number. Therefore, 2 is a factor of 84.

• Divide the result (42) by 2. If the result is a whole number, then 2 is a factor of the number.

For 42, 42 / 2 = 21, which is a whole number. Therefore, 2 is a factor of 42.

• Divide the result (21) by 2. If the result is a whole number, then 2 is a factor of the number.

For 21, 21 / 2 = 10.5, which is not a whole number. Therefore, 2 is not a factor of 21.

• Divide the result (21) by the next smallest prime number (in this case, 3). If the result is a whole number, then 3 is a factor of the number.

For 21, 21 / 3 = 7, which is a whole number. Therefore, 3 is a factor of 21.

• Divide the result (7) by the next smallest prime number (in this case, 5). If the result is a whole number, then 5 is a factor of the number.

For 7, 7 / 5 is not a whole number. Therefore, 5 is not a factor of 7.

Factor Pairs of 84

A factor pair of a number is a pair of numbers that can be multiplied together to get the original number. Here are the factor pairs of 84:

1 x 84

2 x 42

3 x 28

4 x 21

6 x 14

7 x 12

To find the factor pairs of a number, we can simply list out all the numbers that can be multiplied together to get the original number. For example, to find the factor pairs of 84, we can list out all the numbers that can be multiplied together to get 84:

1 x 84 = 84

2 x 42 = 84

3 x 28 = 84

4 x 21 = 84

6 x 14 = 84

7 x 12 = 84

More Factors

• Factors of 81
• Factors of 82
• Factors of 83
• Factors of 84
• Factors of 86
• Factors of 87
• Factors of 88

Factors of 84 – Quick Recap

• Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
• Negative Factors of 84: -1, -2, -3, -4, -6, -7, -12, -14, -21, -28, -42 and -84.
• Prime Factors of 84:  2, 3, 7.
• Prime Factorization of 84: 2 × 2 × 3 × 7.

Factors of 84 – Fun Facts

• 84 is a composite number, which means that it can be written as the product of two or more smaller numbers. In this case, the prime factorization of 84 is 2 x 2 x 3 x 7, which means that the factors of 84 include all the combinations of these prime numbers.
• The number 84 has 12 factors in total. These include 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.
• The factors of 84 can be split into two groups: the factors of 84 that are less than 84 (also known as the “proper factors” of 84), and the factors of 84 that are equal to 84 (also known as the “improper factors” of 84). The proper factors of 84 are 1, 2, 3, 4, 6, 7, 12, 14, 21, and 28, while the improper factors of 84 are 84 itself and 42 (which is half of 84).
• 84 is a “highly composite number,” which means that it has more factors than any smaller number.
• 84 is the double factorial of 7, which means that it is the product of all the integers from 1 to 7, with each integer being multiplied by the next smaller integer. For example, 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040, and 84 = 2 x 7! = 2 x 5040 = 10080.

Also Check: Multiples, Square Root, and LCM

Solved Examples of Factor of 84

Q.1: What is the smallest number that can go into 84?
Solution: The smallest number that can go into 84 is 1 since any number divided by 1 will remain unchanged.

Q.2: Find the prime factorization of 84.
Solution: The prime factorization of 84 is 2 x 2 x 3 x 7, since 2 x 2 = 4, 3 x 7 = 21, and 4 x 21 = 84.

Q.3: How many even factors does 84 have?
Solution: There are 8 even factors of 84; 2, 4, 6,12,14, 28, 42, and 84.

Q.4: List all the pairs of factors for 84 in increasing order.
Solution: The pairs of factors for 72 in increasing order are (1,84), (2,42), (3,28), (4,21), (6,14), and (7,12).

Q.5: What is the greatest common factor for 79 and 83?
Solution: The greatest common factor for 79 and 83 is 1 since no other integer between them can be evenly divided into both numbers without a remainder ​(​no shared divisors)​ .​

Q.6: Is there any perfect square within the range of numerators from 81 to 85?
Solution: The square root of 81 is 9, which is an integer. Therefore, 81 is a perfect square. The square root of 82 is not an integer. The square root of 83 is not an integer. The square root of 84 is not an integer. The square root of 85 is not an integer. So, within the range of numerators from 81 to 85, only 81 is a perfect square.

Q.7: What would be the least common multiple between 73 and 89?
Solution: The LCM of 73 and 89 is 6497.

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Frequently Asked Questions on Factors of 84