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GCF of 10, 30 and 45

GCF of 10, 30 and 45 is the largest possible number that divides 10, 30 and 45 exactly without any remainder. The factors of 10, 30 and 45 are (1, 2, 5, 10), (1, 2, 3, 5, 6, 10, 15, 30) and (1, 3, 5, 9, 15, 45) respectively. There are 3 commonly used methods to find the GCF of 10, 30 and 45 - Euclidean algorithm, prime factorization, and long division.

1.	GCF of 10, 30 and 45
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 10, 30 and 45?

Answer: GCF of 10, 30 and 45 is 5.

Explanation:

The GCF of three non-zero integers, x(10), y(30) and z(45), is the greatest positive integer m(5) that divides x(10), y(30) and z(45) without any remainder.

Methods to Find GCF of 10, 30 and 45

Let's look at the different methods for finding the GCF of 10, 30 and 45.

Long Division Method
Using Euclid's Algorithm
Listing Common Factors

GCF of 10, 30 and 45 by Long Division

GCF of 10, 30 and 45 can be represented as GCF of (GCF of 10, 30) and 45. GCF(10, 30, 45) can be thus calculated by first finding GCF(10, 30) using long division and thereafter using this result with 45 to perform long division again.

Step 1: Divide 30 (larger number) by 10 (smaller number).
Step 2: Since the remainder = 0, the divisor (10) is the GCF(10, 30) = 10.
Step 3: Now to find the GCF of 10 and 45, we will perform a long division on 45 and 10.
Step 4: For remainder = 0, divisor = 5 ⇒ GCF(10, 45) = 5

Thus, GCF(10, 30, 45) = GCF(GCF(10, 30), 45) = 5.

GCF of 10, 30 and 45 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.

GCF(10, 30, 45) = GCF(GCF(10, 30), 45)

GCF(30, 10) = GCF(10, 30 mod 10) = GCF(10, 0)
GCF(10, 0) = 10 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Steps for GCF(10, 45)

GCF(45, 10) = GCF(10, 45 mod 10) = GCF(10, 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 10, 30 and 45 is 5.

GCF of 10, 30 and 45 by Listing Common Factors

Factors of 10: 1, 2, 5, 10
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 45: 1, 3, 5, 9, 15, 45

There are 2 common factors of 10, 30 and 45, that are 1 and 5. Therefore, the greatest common factor of 10, 30 and 45 is 5.

☛ Also Check:

GCF of 10, 30 and 45 Examples

Example 1: Find the greatest number that divides 10, 30, and 45 completely.

Solution:

The greatest number that divides 10, 30, and 45 exactly is their greatest common factor.
- Factors of 10 = 1, 2, 5, 10
- Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30
- Factors of 45 = 1, 3, 5, 9, 15, 45
The GCF of 10, 30, and 45 is 5.
∴ The greatest number that divides 10, 30, and 45 is 5.
Example 2: Calculate the GCF of 10, 30, and 45 using LCM of the given numbers.

Solution:

Prime factorization of 10, 30 and 45 is given as,
- 10 = 2 × 5
- 30 = 2 × 3 × 5
- 45 = 3 × 3 × 5
LCM(10, 30) = 30, LCM(30, 45) = 90, LCM(45, 10) = 90, LCM(10, 30, 45) = 90
⇒ GCF(10, 30, 45) = [(10 × 30 × 45) × LCM(10, 30, 45)]/[LCM(10, 30) × LCM (30, 45) × LCM(45, 10)]
⇒ GCF(10, 30, 45) = (13500 × 90)/(30 × 90 × 90)
⇒ GCF(10, 30, 45) = 5.
Therefore, the GCF of 10, 30 and 45 is 5.
Example 3: Verify the relation between the LCM and GCF of 10, 30 and 45.

Solution:

The relation between the LCM and GCF of 10, 30 and 45 is given as, GCF(10, 30, 45) = [(10 × 30 × 45) × LCM(10, 30, 45)]/[LCM(10, 30) × LCM (30, 45) × LCM(10, 45)]
⇒ Prime factorization of 10, 30 and 45:
- 10 = 2 × 5
- 30 = 2 × 3 × 5
- 45 = 3 × 3 × 5
∴ LCM of (10, 30), (30, 45), (10, 45), and (10, 30, 45) is 30, 90, 90, and 90 respectively.
Now, LHS = GCF(10, 30, 45) = 5.
And, RHS = [(10 × 30 × 45) × LCM(10, 30, 45)]/[LCM(10, 30) × LCM (30, 45) × LCM(10, 45)] = [(13500) × 90]/[30 × 90 × 90]
LHS = RHS = 5.
Hence verified.

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FAQs on GCF of 10, 30 and 45

What is the GCF of 10, 30 and 45?

The GCF of 10, 30 and 45 is 5. To calculate the greatest common factor (GCF) of 10, 30 and 45, we need to factor each number (factors of 10 = 1, 2, 5, 10; factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30; factors of 45 = 1, 3, 5, 9, 15, 45) and choose the greatest factor that exactly divides 10, 30 and 45, i.e., 5.

What are the Methods to Find GCF of 10, 30 and 45?

There are three commonly used methods to find the GCF of 10, 30 and 45.

By Long Division
By Prime Factorization
By Listing Common Factors

What is the Relation Between LCM and GCF of 10, 30 and 45?

The following equation can be used to express the relation between LCM (Least Common Multiple) and GCF of 10, 30 and 45, i.e. GCF(10, 30, 45) = [(10 × 30 × 45) × LCM(10, 30, 45)]/[LCM(10, 30) × LCM (30, 45) × LCM(10, 45)].
☛ Greatest Common Factor Calculator

Which of the following is GCF of 10, 30 and 45? 5, 69, 49, 65, 55, 93, 53, 47, 59

GCF of 10, 30, 45 will be the number that divides 10, 30, and 45 without leaving any remainder. The only number that satisfies the given condition is 5.

How to Find the GCF of 10, 30 and 45 by Prime Factorization?

To find the GCF of 10, 30 and 45, we will find the prime factorization of given numbers, i.e. 10 = 2 × 5; 30 = 2 × 3 × 5; 45 = 3 × 3 × 5.
⇒ Since 5 is the only common prime factor of 10, 30 and 45. Hence, GCF(10, 30, 45) = 5.
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