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GCF of 32, 56 and 96

GCF of 32, 56 and 96 is the largest possible number that divides 32, 56 and 96 exactly without any remainder. The factors of 32, 56 and 96 are (1, 2, 4, 8, 16, 32), (1, 2, 4, 7, 8, 14, 28, 56) and (1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96) respectively. There are 3 commonly used methods to find the GCF of 32, 56 and 96 - Euclidean algorithm, prime factorization, and long division.

1.	GCF of 32, 56 and 96
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is GCF of 32, 56 and 96?

Answer: GCF of 32, 56 and 96 is 8.

Explanation:

The GCF of three non-zero integers, x(32), y(56) and z(96), is the greatest positive integer m(8) that divides x(32), y(56) and z(96) without any remainder.

Methods to Find GCF of 32, 56 and 96

Let's look at the different methods for finding the GCF of 32, 56 and 96.

Long Division Method
Using Euclid's Algorithm
Listing Common Factors

GCF of 32, 56 and 96 by Long Division

GCF of 32, 56 and 96 can be represented as GCF of (GCF of 32, 56) and 96. GCF(32, 56, 96) can be thus calculated by first finding GCF(32, 56) using long division and thereafter using this result with 96 to perform long division again.

Step 1: Divide 56 (larger number) by 32 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (32) by the remainder (24). Repeat this process until the remainder = 0.
⇒ GCF(32, 56) = 8.
Step 3: Now to find the GCF of 8 and 96, we will perform a long division on 96 and 8.
Step 4: For remainder = 0, divisor = 8 ⇒ GCF(8, 96) = 8

Thus, GCF(32, 56, 96) = GCF(GCF(32, 56), 96) = 8.

GCF of 32, 56 and 96 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(32, 56, 96) = GCF(GCF(32, 56), 96)

GCF(56, 32) = GCF(32, 56 mod 32) = GCF(32, 24)
GCF(32, 24) = GCF(24, 32 mod 24) = GCF(24, 8)
GCF(24, 8) = GCF(8, 24 mod 8) = GCF(8, 0)
GCF(8, 0) = 8 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Steps for GCF(8, 96)

GCF(96, 8) = GCF(8, 96 mod 8) = GCF(8, 0)
GCF(8, 0) = 8 (∵ GCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of GCF of 32, 56 and 96 is 8.

GCF of 32, 56 and 96 by Listing Common Factors

Factors of 32: 1, 2, 4, 8, 16, 32
Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
Factors of 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96

There are 4 common factors of 32, 56 and 96, that are 8, 1, 2, and 4. Therefore, the greatest common factor of 32, 56 and 96 is 8.

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GCF of 32, 56 and 96 Examples

Example 1: Find the greatest number that divides 32, 56, and 96 completely.

Solution:

The greatest number that divides 32, 56, and 96 exactly is their greatest common factor.
- Factors of 32 = 1, 2, 4, 8, 16, 32
- Factors of 56 = 1, 2, 4, 7, 8, 14, 28, 56
- Factors of 96 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
The GCF of 32, 56, and 96 is 8.
∴ The greatest number that divides 32, 56, and 96 is 8.
Example 2: Calculate the GCF of 32, 56, and 96 using LCM of the given numbers.

Solution:

Prime factorization of 32, 56 and 96 is given as,
- 32 = 2 × 2 × 2 × 2 × 2
- 56 = 2 × 2 × 2 × 7
- 96 = 2 × 2 × 2 × 2 × 2 × 3
LCM(32, 56) = 224, LCM(56, 96) = 672, LCM(96, 32) = 96, LCM(32, 56, 96) = 672
⇒ GCF(32, 56, 96) = [(32 × 56 × 96) × LCM(32, 56, 96)]/[LCM(32, 56) × LCM (56, 96) × LCM(96, 32)]
⇒ GCF(32, 56, 96) = (172032 × 672)/(224 × 672 × 96)
⇒ GCF(32, 56, 96) = 8.
Therefore, the GCF of 32, 56 and 96 is 8.
Example 3: Verify the relation between the LCM and GCF of 32, 56 and 96.

Solution:

The relation between the LCM and GCF of 32, 56 and 96 is given as, GCF(32, 56, 96) = [(32 × 56 × 96) × LCM(32, 56, 96)]/[LCM(32, 56) × LCM (56, 96) × LCM(32, 96)]
⇒ Prime factorization of 32, 56 and 96:
- 32 = 2 × 2 × 2 × 2 × 2
- 56 = 2 × 2 × 2 × 7
- 96 = 2 × 2 × 2 × 2 × 2 × 3
∴ LCM of (32, 56), (56, 96), (32, 96), and (32, 56, 96) is 224, 672, 96, and 672 respectively.
Now, LHS = GCF(32, 56, 96) = 8.
And, RHS = [(32 × 56 × 96) × LCM(32, 56, 96)]/[LCM(32, 56) × LCM (56, 96) × LCM(32, 96)] = [(172032) × 672]/[224 × 672 × 96]
LHS = RHS = 8.
Hence verified.

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FAQs on GCF of 32, 56 and 96

What is the GCF of 32, 56 and 96?

The GCF of 32, 56 and 96 is 8. To calculate the GCF of 32, 56 and 96, we need to factor each number (factors of 32 = 1, 2, 4, 8, 16, 32; factors of 56 = 1, 2, 4, 7, 8, 14, 28, 56; factors of 96 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96) and choose the greatest factor that exactly divides 32, 56 and 96, i.e., 8.

Which of the following is GCF of 32, 56 and 96? 8, 130, 140, 115, 118

GCF of 32, 56, 96 will be the number that divides 32, 56, and 96 without leaving any remainder. The only number that satisfies the given condition is 8.

What is the Relation Between LCM and GCF of 32, 56 and 96?

The following equation can be used to express the relation between LCM and GCF of 32, 56 and 96, i.e. GCF(32, 56, 96) = [(32 × 56 × 96) × LCM(32, 56, 96)]/[LCM(32, 56) × LCM (56, 96) × LCM(32, 96)].
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What are the Methods to Find GCF of 32, 56 and 96?

There are three commonly used methods to find the GCF of 32, 56 and 96.

By Long Division
By Prime Factorization
By Listing Common Factors

How to Find the GCF of 32, 56 and 96 by Prime Factorization?

To find the GCF of 32, 56 and 96, we will find the prime factorization of given numbers, i.e. 32 = 2 × 2 × 2 × 2 × 2; 56 = 2 × 2 × 2 × 7; 96 = 2 × 2 × 2 × 2 × 2 × 3.
⇒ Since 2, 2, 2 are common terms in the prime factorization of 32, 56 and 96. Hence, GCF(32, 56, 96) = 2 × 2 × 2 = 8
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