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GCF of 63 and 105

GCF of 63 and 105 is the largest possible number that divides 63 and 105 exactly without any remainder. The factors of 63 and 105 are 1, 3, 7, 9, 21, 63 and 1, 3, 5, 7, 15, 21, 35, 105 respectively. There are 3 commonly used methods to find the GCF of 63 and 105 - prime factorization, Euclidean algorithm, and long division.

1.	GCF of 63 and 105
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 63 and 105?

Answer: GCF of 63 and 105 is 21.

Explanation:

The GCF of two non-zero integers, x(63) and y(105), is the greatest positive integer m(21) that divides both x(63) and y(105) without any remainder.

Methods to Find GCF of 63 and 105

Let's look at the different methods for finding the GCF of 63 and 105.

Using Euclid's Algorithm
Prime Factorization Method
Long Division Method

GCF of 63 and 105 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 = 105 and Y = 63

GCF(105, 63) = GCF(63, 105 mod 63) = GCF(63, 42)
GCF(63, 42) = GCF(42, 63 mod 42) = GCF(42, 21)
GCF(42, 21) = GCF(21, 42 mod 21) = GCF(21, 0)
GCF(21, 0) = 21 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 63 and 105 is 21.

GCF of 63 and 105 by Prime Factorization

Prime factorization of 63 and 105 is (3 × 3 × 7) and (3 × 5 × 7) respectively. As visible, 63 and 105 have common prime factors. Hence, the GCF of 63 and 105 is 3 × 7 = 21.

GCF of 63 and 105 by Long Division

GCF of 63 and 105 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 105 (larger number) by 63 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (63) by the remainder (42).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (21) is the GCF of 63 and 105.

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GCF of 63 and 105 Examples

Example 1: The product of two numbers is 6615. If their GCF is 21, what is their LCM?

Solution:

Given: GCF = 21 and product of numbers = 6615
∵ LCM × GCF = product of numbers
⇒ LCM = Product/GCF = 6615/21
Therefore, the LCM is 315.
Example 2: For two numbers, GCF = 21 and LCM = 315. If one number is 63, find the other number.

Solution:

Given: GCF (x, 63) = 21 and LCM (x, 63) = 315
∵ GCF × LCM = 63 × (x)
⇒ x = (GCF × LCM)/63
⇒ x = (21 × 315)/63
⇒ x = 105
Therefore, the other number is 105.
Example 3: Find the greatest number that divides 63 and 105 exactly.

Solution:

The greatest number that divides 63 and 105 exactly is their greatest common factor, i.e. GCF of 63 and 105.
⇒ Factors of 63 and 105:
- Factors of 63 = 1, 3, 7, 9, 21, 63
- Factors of 105 = 1, 3, 5, 7, 15, 21, 35, 105
Therefore, the GCF of 63 and 105 is 21.

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FAQs on GCF of 63 and 105

What is the GCF of 63 and 105?

The GCF of 63 and 105 is 21. To calculate the GCF (Greatest Common Factor) of 63 and 105, we need to factor each number (factors of 63 = 1, 3, 7, 9, 21, 63; factors of 105 = 1, 3, 5, 7, 15, 21, 35, 105) and choose the greatest factor that exactly divides both 63 and 105, i.e., 21.

What are the Methods to Find GCF of 63 and 105?

There are three commonly used methods to find the GCF of 63 and 105.

By Listing Common Factors
By Long Division
By Prime Factorization

How to Find the GCF of 63 and 105 by Long Division Method?

To find the GCF of 63, 105 using long division method, 105 is divided by 63. The corresponding divisor (21) when remainder equals 0 is taken as GCF.

If the GCF of 105 and 63 is 21, Find its LCM.

GCF(105, 63) × LCM(105, 63) = 105 × 63
Since the GCF of 105 and 63 = 21
⇒ 21 × LCM(105, 63) = 6615
Therefore, LCM = 315
☛ Greatest Common Factor Calculator

What is the Relation Between LCM and GCF of 63, 105?

The following equation can be used to express the relation between Least Common Multiple (LCM) and GCF of 63 and 105, i.e. GCF × LCM = 63 × 105.

How to Find the GCF of 63 and 105 by Prime Factorization?

To find the GCF of 63 and 105, we will find the prime factorization of the given numbers, i.e. 63 = 3 × 3 × 7; 105 = 3 × 5 × 7.
⇒ Since 3, 7 are common terms in the prime factorization of 63 and 105. Hence, GCF(63, 105) = 3 × 7 = 21
☛ What are Prime Numbers?

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