GCF of 36 and 84
2026-02-28 08:23 Diff
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Last updated on September 11, 2025

The GCF is the largest number that can divide two or more numbers without leaving any remainder. GCF is used to share items equally, to group or arrange items, and schedule events. In this topic, we will learn about the GCF of 36 and 84.

What is the GCF of 36 and 84?

The greatest common factor of 36 and 84 is 12. The largest divisor of two or more numbers is called the GCF of the number. If two numbers are co-prime, they have no common factors other than 1, so their GCF is 1. The GCF of two numbers cannot be negative because divisors are always positive.

How to find the GCF of 36 and 84?

To find the GCF of 36 and 84, a few methods are described below -

  1. Listing Factors
  2. Prime Factorization
  3. Long Division Method / by Euclidean Algorithm

GCF of 36 and 84 by Using Listing of Factors

Steps to find the GCF of 36 and 84 using the listing of factors

Step 1: Firstly, list the factors of each number

Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36.

Factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.

Step 2: Now, identify the common factors of them Common factors of 36 and 84: 1, 2, 3, 4, 6, 12.

Step 3: Choose the largest factor The largest factor that both numbers have is 12. The GCF of 36 and 84 is 12.

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GCF of 36 and 84 Using Prime Factorization

To find the GCF of 36 and 84 using Prime Factorization Method, follow these steps:

Step 1: Find the prime factors of each number

Prime Factors of 36: 36 = 2 x 2 x 3 x 3 = 2² x 3²

Prime Factors of 84: 84 = 2 x 2 x 3 x 7 = 2² x 3 x 7

Step 2: Now, identify the common prime factors The common prime factors are: 2 x 2 x 3 = 2² x 3

Step 3: Multiply the common prime factors 2² x 3 = 4 x 3 = 12.

The Greatest Common Factor of 36 and 84 is 12.

GCF of 36 and 84 Using Division Method or Euclidean Algorithm Method

Find the GCF of 36 and 84 using the division method or Euclidean Algorithm Method. Follow these steps:

Step 1: First, divide the larger number by the smaller number Here, divide 84 by 36 84 ÷ 36 = 2 (quotient), The remainder is calculated as 84 − (36 x 2) = 12

The remainder is 12, not zero, so continue the process

Step 2: Now divide the previous divisor (36) by the previous remainder (12) Divide 36 by 12 36 ÷ 12 = 3 (quotient), remainder = 36 − (12 x 3) = 0

The remainder is zero, the divisor will become the GCF. The GCF of 36 and 84 is 12.

Common Mistakes and How to Avoid Them in GCF of 36 and 84

Finding GCF of 36 and 84 looks simple, but students often make mistakes while calculating the GCF. Here are some common mistakes to be avoided by the students.

Problem 1

A chef has 36 apples and 84 oranges. She wants to distribute them into equal fruit baskets, with the largest number of fruits in each basket. How many fruits will be in each basket?

Okay, lets begin

We should find the GCF of 36 and 84 GCF of 36 and 84 2² x 3 = 4 x 3 = 12.

There are 12 equal fruit baskets 36 ÷ 12 = 3 84 ÷ 12 = 7

There will be 12 baskets, and each basket gets 3 apples and 7 oranges.

Explanation

As the GCF of 36 and 84 is 12, the chef can make 12 baskets. Now divide 36 and 84 by 12. Each basket gets 3 apples and 7 oranges.

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Problem 2

A park has 36 benches and 84 trees. They want to arrange them in rows with the same number of benches and trees in each row, using the largest possible number of benches and trees per row. How many benches and trees will be in each row?

Okay, lets begin

GCF of 36 and 84 2² x 3 = 4 x 3 = 12.

So each row will have 12 benches and 12 trees.

Explanation

There are 36 benches and 84 trees. To find the total number of benches and trees in each row, we should find the GCF of 36 and 84. There will be 12 benches and trees in each row.

Well explained 👍

Problem 3

A factory has 36 meters of red wire and 84 meters of blue wire. They want to cut both wires into pieces of equal length, using the longest possible length. What should be the length of each piece?

Okay, lets begin

For calculating the longest equal length, we have to calculate the GCF of 36 and 84 The GCF of 36 and 84 2² x 3 = 4 x 3 = 12. The wire is 12 meters long.

Explanation

For calculating the longest length of the wire, first, we need to calculate the GCF of 36 and 84, which is 12. The length of each piece of the wire will be 12 meters.

Well explained 👍

Problem 4

A construction site has two wooden beams, one 36 cm long and the other 84 cm long. They want to cut them into the longest possible equal pieces, without any wood left over. What should be the length of each piece?

Okay, lets begin

The site needs the longest piece of wood GCF of 36 and 84 2² x 3 = 4 x 3 = 12. The longest length of each piece is 12 cm.

Explanation

To find the longest length of each piece of the two wooden beams, 36 cm and 84 cm, respectively. We have to find the GCF of 36 and 84, which is 12 cm. The longest length of each piece is 12 cm.

Well explained 👍

Problem 5

If the GCF of 36 and ‘b’ is 12, and the LCM is 252. Find ‘b’.

Okay, lets begin

The value of ‘b’ is 84.

Explanation

GCF x LCM = product of the numbers

12 x 252 = 36 x b

3024 = 36b

b = 3024 ÷ 36 = 84

Well explained 👍

FAQs on the Greatest Common Factor of 36 and 84

1.What is the LCM of 36 and 84?

The LCM of 36 and 84 is 252.

2.Is 36 divisible by 2?

Yes, 36 is divisible by 2 because it is an even number.

3.What will be the GCF of any two prime numbers?

The common factor of prime numbers is 1 and the number itself. Since 1 is the only common factor of any two prime numbers, it is said to be the GCF of any two prime numbers.

4.What is the prime factorization of 84?

The prime factorization of 84 is 2² x 3 x 7.

5.Are 36 and 84 prime numbers?

No, 36 and 84 are not prime numbers because both of them have more than two factors.

Important Glossaries for GCF of 36 and 84

  • Factors: Factors are numbers that divide the target number completely. For example, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
  • Multiple: Multiples are the products we get by multiplying a given number by another. For example, the multiples of 4 are 4, 8, 12, 16, 20, and so on.
  • Prime Factors: These are the factors of a number that are prime numbers and divide the given number completely. For example, the prime factors of 84 are 2, 3, and 7.
  • Remainder: The value left after division when the number cannot be divided evenly. For example, when 12 is divided by 7, the remainder is 5, and the quotient is 1.
  • LCM: The smallest common multiple of two or more numbers is termed LCM. For example, the LCM of 36 and 84 is 252.
  • GCF: The largest factor that commonly divides two or more numbers. For example, the GCF of 36 and 84 is 12, as it is their largest common factor that divides the numbers completely.

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Hiralee Lalitkumar Makwana

About the Author

Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.

Fun Fact

: She loves to read number jokes and games.