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GCF of 52 and 78

GCF of 52 and 78 is the largest possible number that divides 52 and 78 exactly without any remainder. The factors of 52 and 78 are 1, 2, 4, 13, 26, 52 and 1, 2, 3, 6, 13, 26, 39, 78 respectively. There are 3 commonly used methods to find the GCF of 52 and 78 - long division, Euclidean algorithm, and prime factorization.

1.	GCF of 52 and 78
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 52 and 78?

Answer: GCF of 52 and 78 is 26.

Explanation:

The GCF of two non-zero integers, x(52) and y(78), is the greatest positive integer m(26) that divides both x(52) and y(78) without any remainder.

Methods to Find GCF of 52 and 78

Let's look at the different methods for finding the GCF of 52 and 78.

Long Division Method
Using Euclid's Algorithm
Prime Factorization Method

GCF of 52 and 78 by Long Division

GCF of 52 and 78 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 78 (larger number) by 52 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (52) by the remainder (26).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (26) is the GCF of 52 and 78.

GCF of 52 and 78 by Euclidean Algorithm

As per the Euclidean Algorithm, GCF(X, Y) = GCF(Y, X mod Y)
where X > Y and mod is the modulo operator.

Here X = 78 and Y = 52

GCF(78, 52) = GCF(52, 78 mod 52) = GCF(52, 26)
GCF(52, 26) = GCF(26, 52 mod 26) = GCF(26, 0)
GCF(26, 0) = 26 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 52 and 78 is 26.

GCF of 52 and 78 by Prime Factorization

Prime factorization of 52 and 78 is (2 × 2 × 13) and (2 × 3 × 13) respectively. As visible, 52 and 78 have common prime factors. Hence, the GCF of 52 and 78 is 2 × 13 = 26.

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GCF of 52 and 78 Examples

Example 1: The product of two numbers is 4056. If their GCF is 26, what is their LCM?

Solution:

Given: GCF = 26 and product of numbers = 4056
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 4056/26
Therefore, the LCM is 156.
Example 2: Find the greatest number that divides 52 and 78 exactly.

Solution:

The greatest number that divides 52 and 78 exactly is their greatest common factor, i.e. GCF of 52 and 78.
⇒ Factors of 52 and 78:
- Factors of 52 = 1, 2, 4, 13, 26, 52
- Factors of 78 = 1, 2, 3, 6, 13, 26, 39, 78
Therefore, the GCF of 52 and 78 is 26.
Example 3: Find the GCF of 52 and 78, if their LCM is 156.

Solution:

∵ LCM × GCF = 52 × 78
⇒ GCF(52, 78) = (52 × 78)/156 = 26
Therefore, the greatest common factor of 52 and 78 is 26.

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FAQs on GCF of 52 and 78

What is the GCF of 52 and 78?

The GCF of 52 and 78 is 26. To calculate the GCF of 52 and 78, we need to factor each number (factors of 52 = 1, 2, 4, 13, 26, 52; factors of 78 = 1, 2, 3, 6, 13, 26, 39, 78) and choose the greatest factor that exactly divides both 52 and 78, i.e., 26.

If the GCF of 78 and 52 is 26, Find its LCM.

GCF(78, 52) × LCM(78, 52) = 78 × 52
Since the GCF of 78 and 52 = 26
⇒ 26 × LCM(78, 52) = 4056
Therefore, LCM = 156
☛ Greatest Common Factor Calculator

How to Find the GCF of 52 and 78 by Long Division Method?

To find the GCF of 52, 78 using long division method, 78 is divided by 52. The corresponding divisor (26) when remainder equals 0 is taken as GCF.

What is the Relation Between LCM and GCF of 52, 78?

The following equation can be used to express the relation between Least Common Multiple (LCM) and GCF of 52 and 78, i.e. GCF × LCM = 52 × 78.

How to Find the GCF of 52 and 78 by Prime Factorization?

To find the GCF of 52 and 78, we will find the prime factorization of the given numbers, i.e. 52 = 2 × 2 × 13; 78 = 2 × 3 × 13.
⇒ Since 2, 13 are common terms in the prime factorization of 52 and 78. Hence, GCF(52, 78) = 2 × 13 = 26
☛ What is a Prime Number?

What are the Methods to Find GCF of 52 and 78?

There are three commonly used methods to find the GCF of 52 and 78.

By Long Division
By Listing Common Factors
By Prime Factorization

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