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GCF of 78 and 104

GCF of 78 and 104 is the largest possible number that divides 78 and 104 exactly without any remainder. The factors of 78 and 104 are 1, 2, 3, 6, 13, 26, 39, 78 and 1, 2, 4, 8, 13, 26, 52, 104 respectively. There are 3 commonly used methods to find the GCF of 78 and 104 - long division, prime factorization, and Euclidean algorithm.

1.	GCF of 78 and 104
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 78 and 104?

Answer: GCF of 78 and 104 is 26.

Explanation:

The GCF of two non-zero integers, x(78) and y(104), is the greatest positive integer m(26) that divides both x(78) and y(104) without any remainder.

Methods to Find GCF of 78 and 104

The methods to find the GCF of 78 and 104 are explained below.

Using Euclid's Algorithm
Prime Factorization Method
Long Division Method

GCF of 78 and 104 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 = 104 and Y = 78

GCF(104, 78) = GCF(78, 104 mod 78) = GCF(78, 26)
GCF(78, 26) = GCF(26, 78 mod 26) = GCF(26, 0)
GCF(26, 0) = 26 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 78 and 104 is 26.

GCF of 78 and 104 by Prime Factorization

Prime factorization of 78 and 104 is (2 × 3 × 13) and (2 × 2 × 2 × 13) respectively. As visible, 78 and 104 have common prime factors. Hence, the GCF of 78 and 104 is 2 × 13 = 26.

GCF of 78 and 104 by Long Division

GCF of 78 and 104 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 104 (larger number) by 78 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (78) by the remainder (26).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (26) is the GCF of 78 and 104.

☛ Also Check:

GCF of 78 and 104 Examples

Example 1: Find the GCF of 78 and 104, if their LCM is 312.

Solution:

∵ LCM × GCF = 78 × 104
⇒ GCF(78, 104) = (78 × 104)/312 = 26
Therefore, the greatest common factor of 78 and 104 is 26.
Example 2: Find the greatest number that divides 78 and 104 exactly.

Solution:

The greatest number that divides 78 and 104 exactly is their greatest common factor, i.e. GCF of 78 and 104.
⇒ Factors of 78 and 104:
- Factors of 78 = 1, 2, 3, 6, 13, 26, 39, 78
- Factors of 104 = 1, 2, 4, 8, 13, 26, 52, 104
Therefore, the GCF of 78 and 104 is 26.
Example 3: For two numbers, GCF = 26 and LCM = 312. If one number is 104, find the other number.

Solution:

Given: GCF (y, 104) = 26 and LCM (y, 104) = 312
∵ GCF × LCM = 104 × (y)
⇒ y = (GCF × LCM)/104
⇒ y = (26 × 312)/104
⇒ y = 78
Therefore, the other number is 78.

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FAQs on GCF of 78 and 104

What is the GCF of 78 and 104?

The GCF of 78 and 104 is 26. To calculate the greatest common factor of 78 and 104, we need to factor each number (factors of 78 = 1, 2, 3, 6, 13, 26, 39, 78; factors of 104 = 1, 2, 4, 8, 13, 26, 52, 104) and choose the greatest factor that exactly divides both 78 and 104, i.e., 26.

How to Find the GCF of 78 and 104 by Prime Factorization?

To find the GCF of 78 and 104, we will find the prime factorization of the given numbers, i.e. 78 = 2 × 3 × 13; 104 = 2 × 2 × 2 × 13.
⇒ Since 2, 13 are common terms in the prime factorization of 78 and 104. Hence, GCF(78, 104) = 2 × 13 = 26
☛ Prime Number

If the GCF of 104 and 78 is 26, Find its LCM.

GCF(104, 78) × LCM(104, 78) = 104 × 78
Since the GCF of 104 and 78 = 26
⇒ 26 × LCM(104, 78) = 8112
Therefore, LCM = 312
☛ Greatest Common Factor Calculator

What are the Methods to Find GCF of 78 and 104?

There are three commonly used methods to find the GCF of 78 and 104.

By Long Division
By Prime Factorization
By Listing Common Factors

What is the Relation Between LCM and GCF of 78, 104?

The following equation can be used to express the relation between LCM (Least Common Multiple) and GCF of 78 and 104, i.e. GCF × LCM = 78 × 104.

How to Find the GCF of 78 and 104 by Long Division Method?

To find the GCF of 78, 104 using long division method, 104 is divided by 78. The corresponding divisor (26) when remainder equals 0 is taken as GCF.

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