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HCF of 54, 288 and 360

HCF of 54, 288 and 360 is the largest possible number that divides 54, 288 and 360 exactly without any remainder. The factors of 54, 288 and 360 are (1, 2, 3, 6, 9, 18, 27, 54), (1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, 288) and (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360) respectively. There are 3 commonly used methods to find the HCF of 54, 288 and 360 - long division, Euclidean algorithm, and prime factorization.

1.	HCF of 54, 288 and 360
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 54, 288 and 360?

Answer: HCF of 54, 288 and 360 is 18.

Explanation:

The HCF of three non-zero integers, x(54), y(288) and z(360), is the highest positive integer m(18) that divides x(54), y(288) and z(360) without any remainder.

Methods to Find HCF of 54, 288 and 360

Let's look at the different methods for finding the HCF of 54, 288 and 360.

Listing Common Factors
Prime Factorization Method
Long Division Method

HCF of 54, 288 and 360 by Listing Common Factors

Factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
Factors of 288: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, 288
Factors of 360: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360

There are 6 common factors of 54, 288 and 360, that are 1, 2, 3, 6, 9, and 18. Therefore, the highest common factor of 54, 288 and 360 is 18.

HCF of 54, 288 and 360 by Prime Factorization

Prime factorization of 54, 288 and 360 is (2 × 3 × 3 × 3), (2 × 2 × 2 × 2 × 2 × 3 × 3) and (2 × 2 × 2 × 3 × 3 × 5) respectively. As visible, 54, 288 and 360 have common prime factors. Hence, the HCF of 54, 288 and 360 is 2 × 3 × 3 = 18.

HCF of 54, 288 and 360 by Long Division

HCF of 54, 288 and 360 can be represented as HCF of (HCF of 54, 288) and 360. HCF(54, 288, 360) can be thus calculated by first finding HCF(54, 288) using long division and thereafter using this result with 360 to perform long division again.

Step 1: Divide 288 (larger number) by 54 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (54) by the remainder (18). Repeat this process until the remainder = 0.
⇒ HCF(54, 288) = 18.
Step 3: Now to find the HCF of 18 and 360, we will perform a long division on 360 and 18.
Step 4: For remainder = 0, divisor = 18 ⇒ HCF(18, 360) = 18

Thus, HCF(54, 288, 360) = HCF(HCF(54, 288), 360) = 18.

☛ Also Check:

HCF of 54, 288 and 360 Examples

Example 1: Calculate the HCF of 54, 288, and 360 using LCM of the given numbers.

Solution:

Prime factorization of 54, 288 and 360 is given as,
- 54 = 2 × 3 × 3 × 3
- 288 = 2 × 2 × 2 × 2 × 2 × 3 × 3
- 360 = 2 × 2 × 2 × 3 × 3 × 5
LCM(54, 288) = 864, LCM(288, 360) = 1440, LCM(360, 54) = 1080, LCM(54, 288, 360) = 4320
⇒ HCF(54, 288, 360) = [(54 × 288 × 360) × LCM(54, 288, 360)]/[LCM(54, 288) × LCM (288, 360) × LCM(360, 54)]
⇒ HCF(54, 288, 360) = (5598720 × 4320)/(864 × 1440 × 1080)
⇒ HCF(54, 288, 360) = 18.
Therefore, the HCF of 54, 288 and 360 is 18.
Example 2: Find the highest number that divides 54, 288, and 360 completely.

Solution:

The highest number that divides 54, 288, and 360 exactly is their highest common factor.
- Factors of 54 = 1, 2, 3, 6, 9, 18, 27, 54
- Factors of 288 = 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, 288
- Factors of 360 = 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
The HCF of 54, 288, and 360 is 18.
∴ The highest number that divides 54, 288, and 360 is 18.
Example 3: Verify the relation between the LCM and HCF of 54, 288 and 360.

Solution:

The relation between the LCM and HCF of 54, 288 and 360 is given as, HCF(54, 288, 360) = [(54 × 288 × 360) × LCM(54, 288, 360)]/[LCM(54, 288) × LCM (288, 360) × LCM(54, 360)]
⇒ Prime factorization of 54, 288 and 360:
- 54 = 2 × 3 × 3 × 3
- 288 = 2 × 2 × 2 × 2 × 2 × 3 × 3
- 360 = 2 × 2 × 2 × 3 × 3 × 5
∴ LCM of (54, 288), (288, 360), (54, 360), and (54, 288, 360) is 864, 1440, 1080, and 4320 respectively.
Now, LHS = HCF(54, 288, 360) = 18.
And, RHS = [(54 × 288 × 360) × LCM(54, 288, 360)]/[LCM(54, 288) × LCM (288, 360) × LCM(54, 360)] = [(5598720) × 4320]/[864 × 1440 × 1080]
LHS = RHS = 18.
Hence verified.

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FAQs on HCF of 54, 288 and 360

What is the HCF of 54, 288 and 360?

The HCF of 54, 288 and 360 is 18. To calculate the HCF of 54, 288 and 360, we need to factor each number (factors of 54 = 1, 2, 3, 6, 9, 18, 27, 54; factors of 288 = 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, 288; factors of 360 = 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360) and choose the highest factor that exactly divides 54, 288 and 360, i.e., 18.

What is the Relation Between LCM and HCF of 54, 288 and 360?

The following equation can be used to express the relation between LCM (Least Common Multiple) and HCF of 54, 288 and 360, i.e. HCF(54, 288, 360) = [(54 × 288 × 360) × LCM(54, 288, 360)]/[LCM(54, 288) × LCM (288, 360) × LCM(54, 360)].
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Which of the following is HCF of 54, 288 and 360? 18, 362, 388, 374, 395, 401, 399, 371, 371

HCF of 54, 288, 360 will be the number that divides 54, 288, and 360 without leaving any remainder. The only number that satisfies the given condition is 18.

What are the Methods to Find HCF of 54, 288 and 360?

There are three commonly used methods to find the HCF of 54, 288 and 360.

By Prime Factorization
By Listing Common Factors
By Long Division

How to Find the HCF of 54, 288 and 360 by Prime Factorization?

To find the HCF of 54, 288 and 360, we will find the prime factorization of given numbers, i.e. 54 = 2 × 3 × 3 × 3; 288 = 2 × 2 × 2 × 2 × 2 × 3 × 3; 360 = 2 × 2 × 2 × 3 × 3 × 5.
⇒ Since 2, 3, 3 are common terms in the prime factorization of 54, 288 and 360. Hence, HCF(54, 288, 360) = 2 × 3 × 3 = 18
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