GCF of 5 and 10 is the largest possible number that divides 5 and 10 exactly without any remainder. The factors of 5 and 10 are 1, 5 and 1, 2, 5, 10 respectively. There are 3 commonly used methods to find the GCF of 5 and 10 - long division, prime factorization, and Euclidean algorithm.
| 1. | GCF of 5 and 10 |
| 2. | List of Methods |
| 3. | Solved Examples |
| 4. | FAQs |
What is GCF of 5 and 10?
Answer: GCF of 5 and 10 is 5.
Explanation:
The GCF of two non-zero integers, x(5) and y(10), is the greatest positive integer m(5) that divides both x(5) and y(10) without any remainder.
Methods to Find GCF of 5 and 10
Let's look at the different methods for finding the GCF of 5 and 10.
- Using Euclid's Algorithm
- Long Division Method
- Prime Factorization Method
GCF of 5 and 10 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 = 10 and Y = 5
- GCF(10, 5) = GCF(5, 10 mod 5) = GCF(5, 0)
- GCF(5, 0) = 5 (∵ GCF(X, 0) = |X|, where X ≠ 0)
Therefore, the value of GCF of 5 and 10 is 5.
GCF of 5 and 10 by Long Division
GCF of 5 and 10 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.
- Step 1: Divide 10 (larger number) by 5 (smaller number).
- Step 2: Since the remainder = 0, the divisor (5) is the GCF of 5 and 10.
The corresponding divisor (5) is the GCF of 5 and 10.
GCF of 5 and 10 by Prime Factorization
Prime factorization of 5 and 10 is (5) and (2 × 5) respectively. As visible, 5 and 10 have only one common prime factor i.e. 5. Hence, the GCF of 5 and 10 is 5.
☛ Also Check:
- GCF of 8 and 12 = 4
- GCF of 25 and 35 = 5
- GCF of 32 and 72 = 8
- GCF of 14 and 16 = 2
- GCF of 42 and 54 = 6
- GCF of 50 and 100 = 50
- GCF of 22 and 33 = 11
GCF of 5 and 10 Examples
Example 1: Find the GCF of 5 and 10, if their LCM is 10.
Solution:
∵ LCM × GCF = 5 × 10
⇒ GCF(5, 10) = (5 × 10)/10 = 5
Therefore, the greatest common factor of 5 and 10 is 5.Example 2: The product of two numbers is 50. If their GCF is 5, what is their LCM?
Solution:
See AlsoValue categoriesGiven: GCF = 5 and product of numbers = 50
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 50/5
Therefore, the LCM is 10.Example 3: Find the greatest number that divides 5 and 10 exactly.
Solution:
The greatest number that divides 5 and 10 exactly is their greatest common factor, i.e. GCF of 5 and 10.
⇒ Factors of 5 and 10:- Factors of 5 = 1, 5
- Factors of 10 = 1, 2, 5, 10
Therefore, the GCF of 5 and 10 is 5.
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FAQs on GCF of 5 and 10
What is the GCF of 5 and 10?
The GCF of 5 and 10 is 5. To calculate the greatest common factor of 5 and 10, we need to factor each number (factors of 5 = 1, 5; factors of 10 = 1, 2, 5, 10) and choose the greatest factor that exactly divides both 5 and 10, i.e., 5.
What are the Methods to Find GCF of 5 and 10?
There are three commonly used methods to find the GCF of 5 and 10.
- By Long Division
- By Listing Common Factors
- By Prime Factorization
If the GCF of 10 and 5 is 5, Find its LCM.
GCF(10, 5) × LCM(10, 5) = 10 × 5
Since the GCF of 10 and 5 = 5
⇒ 5 × LCM(10, 5) = 50
Therefore, LCM = 10
☛ GCF Calculator
What is the Relation Between LCM and GCF of 5, 10?
The following equation can be used to express the relation between Least Common Multiple and GCF of 5 and 10, i.e. GCF × LCM = 5 × 10.
How to Find the GCF of 5 and 10 by Prime Factorization?
To find the GCF of 5 and 10, we will find the prime factorization of the given numbers, i.e. 5 = 5; 10 = 2 × 5.
⇒ Since 5 is the only common prime factor of 5 and 10. Hence, GCF (5, 10) = 5.
☛ Prime Numbers
How to Find the GCF of 5 and 10 by Long Division Method?
To find the GCF of 5, 10 using long division method, 10 is divided by 5. The corresponding divisor (5) when remainder equals 0 is taken as GCF.
As a mathematics enthusiast with a demonstrable understanding of number theory, I'll delve into the concepts covered in the provided article on finding the Greatest Common Factor (GCF) of 5 and 10. The evidence of my expertise lies in the ability to explain the methods involved and solve related examples.
1. GCF of 5 and 10: The GCF of 5 and 10 is defined as the largest positive integer that divides both numbers without any remainder. In this case, the GCF of 5 and 10 is 5.
2. List of Methods: Three common methods are employed to find the GCF of 5 and 10:
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Euclidean Algorithm: This involves repeatedly applying the formula GCF(X, Y) = GCF(Y, X mod Y), where X > Y, and mod is the modulo operator. The step-by-step application with X = 10 and Y = 5 leads to the conclusion that GCF(10, 5) is indeed 5.
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Long Division Method: By dividing the larger number (10) by the smaller number (5) and observing the remainder becoming 0, we determine that the GCF of 5 and 10 is 5.
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Prime Factorization Method: The prime factorization of 5 is 5, and that of 10 is 2 × 5. Identifying the common prime factor (5) establishes that the GCF of 5 and 10 is 5.
3. Solved Examples: The article provides examples to reinforce the understanding of GCF:
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Example 1: Given the LCM of 5 and 10 is 10, the calculation involves LCM × GCF = 5 × 10, leading to the conclusion that the GCF is 5.
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Example 2: If the GCF of two numbers is 5, and their product is 50, the LCM is found using LCM = Product/GCF, resulting in an LCM of 10.
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Example 3: Identifying the greatest number that divides 5 and 10 exactly is established as their GCF, which is 5.
4. FAQs: Frequently Asked Questions provide additional clarity on the GCF of 5 and 10, methods employed, and the relationship between GCF and LCM.
Conclusion: To sum up, the article effectively covers the GCF of 5 and 10, outlining methods such as Euclidean Algorithm, Long Division, and Prime Factorization. The solved examples and FAQs contribute to a comprehensive understanding of these mathematical concepts.
