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GCF of 45 and 72

GCF of 45 and 72 is the largest possible number that divides 45 and 72 exactly without any remainder. The factors of 45 and 72 are 1, 3, 5, 9, 15, 45 and 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 respectively. There are 3 commonly used methods to find the GCF of 45 and 72 - prime factorization, long division, and Euclidean algorithm.

1.	GCF of 45 and 72
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 45 and 72?

Answer: GCF of 45 and 72 is 9.

Explanation:

The GCF of two non-zero integers, x(45) and y(72), is the greatest positive integer m(9) that divides both x(45) and y(72) without any remainder.

Methods to Find GCF of 45 and 72

The methods to find the GCF of 45 and 72 are explained below.

Using Euclid's Algorithm
Long Division Method
Prime Factorization Method

GCF of 45 and 72 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 = 72 and Y = 45

GCF(72, 45) = GCF(45, 72 mod 45) = GCF(45, 27)
GCF(45, 27) = GCF(27, 45 mod 27) = GCF(27, 18)
GCF(27, 18) = GCF(18, 27 mod 18) = GCF(18, 9)
GCF(18, 9) = GCF(9, 18 mod 9) = GCF(9, 0)
GCF(9, 0) = 9 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 45 and 72 is 9.

GCF of 45 and 72 by Long Division

GCF of 45 and 72 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 72 (larger number) by 45 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (45) by the remainder (27).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (9) is the GCF of 45 and 72.

GCF of 45 and 72 by Prime Factorization

Prime factorization of 45 and 72 is (3 × 3 × 5) and (2 × 2 × 2 × 3 × 3) respectively. As visible, 45 and 72 have common prime factors. Hence, the GCF of 45 and 72 is 3 × 3 = 9.

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GCF of 45 and 72 Examples

Example 1: The product of two numbers is 3240. If their GCF is 9, what is their LCM?

Solution:

Given: GCF = 9 and product of numbers = 3240
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 3240/9
Therefore, the LCM is 360.
Example 2: Find the GCF of 45 and 72, if their LCM is 360.

Solution:

∵ LCM × GCF = 45 × 72
⇒ GCF(45, 72) = (45 × 72)/360 = 9
Therefore, the greatest common factor of 45 and 72 is 9.
Example 3: For two numbers, GCF = 9 and LCM = 360. If one number is 72, find the other number.

Solution:

Given: GCF (x, 72) = 9 and LCM (x, 72) = 360
∵ GCF × LCM = 72 × (x)
⇒ x = (GCF × LCM)/72
⇒ x = (9 × 360)/72
⇒ x = 45
Therefore, the other number is 45.

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FAQs on GCF of 45 and 72

What is the GCF of 45 and 72?

The GCF of 45 and 72 is 9. To calculate the greatest common factor (GCF) of 45 and 72, we need to factor each number (factors of 45 = 1, 3, 5, 9, 15, 45; factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72) and choose the greatest factor that exactly divides both 45 and 72, i.e., 9.

What is the Relation Between LCM and GCF of 45, 72?

The following equation can be used to express the relation between LCM (Least Common Multiple) and GCF of 45 and 72, i.e. GCF × LCM = 45 × 72.

How to Find the GCF of 45 and 72 by Long Division Method?

To find the GCF of 45, 72 using long division method, 72 is divided by 45. The corresponding divisor (9) when remainder equals 0 is taken as GCF.

If the GCF of 72 and 45 is 9, Find its LCM.

GCF(72, 45) × LCM(72, 45) = 72 × 45
Since the GCF of 72 and 45 = 9
⇒ 9 × LCM(72, 45) = 3240
Therefore, LCM = 360
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What are the Methods to Find GCF of 45 and 72?

There are three commonly used methods to find the GCF of 45 and 72.

By Long Division
By Listing Common Factors
By Prime Factorization

How to Find the GCF of 45 and 72 by Prime Factorization?

To find the GCF of 45 and 72, we will find the prime factorization of the given numbers, i.e. 45 = 3 × 3 × 5; 72 = 2 × 2 × 2 × 3 × 3.
⇒ Since 3, 3 are common terms in the prime factorization of 45 and 72. Hence, GCF(45, 72) = 3 × 3 = 9
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