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HCF of 120, 144 and 204

HCF of 120, 144 and 204 is the largest possible number that divides 120, 144 and 204 exactly without any remainder. The factors of 120, 144 and 204 are (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120), (1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144) and (1, 2, 3, 4, 6, 12, 17, 34, 51, 68, 102, 204) respectively. There are 3 commonly used methods to find the HCF of 120, 144 and 204 - prime factorization, Euclidean algorithm, and long division.

1.	HCF of 120, 144 and 204
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 120, 144 and 204?

Answer: HCF of 120, 144 and 204 is 12.

Explanation:

The HCF of three non-zero integers, x(120), y(144) and z(204), is the highest positive integer m(12) that divides x(120), y(144) and z(204) without any remainder.

Methods to Find HCF of 120, 144 and 204

The methods to find the HCF of 120, 144 and 204 are explained below.

Using Euclid's Algorithm
Long Division Method
Listing Common Factors

HCF of 120, 144 and 204 by Euclidean Algorithm

As per the Euclidean Algorithm, HCF(X, Y) = HCF(Y, X mod Y)
where X > Y and mod is the modulo operator.

HCF(120, 144, 204) = HCF(HCF(120, 144), 204)

HCF(144, 120) = HCF(120, 144 mod 120) = HCF(120, 24)
HCF(120, 24) = HCF(24, 120 mod 24) = HCF(24, 0)
HCF(24, 0) = 24 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Steps for HCF(24, 204)

HCF(204, 24) = HCF(24, 204 mod 24) = HCF(24, 12)
HCF(24, 12) = HCF(12, 24 mod 12) = HCF(12, 0)
HCF(12, 0) = 12 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of HCF of 120, 144 and 204 is 12.

HCF of 120, 144 and 204 by Long Division

HCF of 120, 144 and 204 can be represented as HCF of (HCF of 120, 144) and 204. HCF(120, 144, 204) can be thus calculated by first finding HCF(120, 144) using long division and thereafter using this result with 204 to perform long division again.

Step 1: Divide 144 (larger number) by 120 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (120) by the remainder (24). Repeat this process until the remainder = 0.
⇒ HCF(120, 144) = 24.
Step 3: Now to find the HCF of 24 and 204, we will perform a long division on 204 and 24.
Step 4: For remainder = 0, divisor = 12 ⇒ HCF(24, 204) = 12

Thus, HCF(120, 144, 204) = HCF(HCF(120, 144), 204) = 12.

HCF of 120, 144 and 204 by Listing Common Factors

Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
Factors of 144: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
Factors of 204: 1, 2, 3, 4, 6, 12, 17, 34, 51, 68, 102, 204

There are 6 common factors of 120, 144 and 204, that are 1, 2, 3, 4, 6, and 12. Therefore, the highest common factor of 120, 144 and 204 is 12.

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HCF of 120, 144 and 204 Examples

Example 1: Verify the relation between the LCM and HCF of 120, 144 and 204.

Solution:

The relation between the LCM and HCF of 120, 144 and 204 is given as, HCF(120, 144, 204) = [(120 × 144 × 204) × LCM(120, 144, 204)]/[LCM(120, 144) × LCM (144, 204) × LCM(120, 204)]
⇒ Prime factorization of 120, 144 and 204:
- 120 = 2 × 2 × 2 × 3 × 5
- 144 = 2 × 2 × 2 × 2 × 3 × 3
- 204 = 2 × 2 × 3 × 17
∴ LCM of (120, 144), (144, 204), (120, 204), and (120, 144, 204) is 720, 2448, 2040, and 12240 respectively.
Now, LHS = HCF(120, 144, 204) = 12.
And, RHS = [(120 × 144 × 204) × LCM(120, 144, 204)]/[LCM(120, 144) × LCM (144, 204) × LCM(120, 204)] = [(3525120) × 12240]/[720 × 2448 × 2040]
LHS = RHS = 12.
Hence verified.
Example 2: Calculate the HCF of 120, 144, and 204 using LCM of the given numbers.

Solution:

Prime factorization of 120, 144 and 204 is given as,
- 120 = 2 × 2 × 2 × 3 × 5
- 144 = 2 × 2 × 2 × 2 × 3 × 3
- 204 = 2 × 2 × 3 × 17
LCM(120, 144) = 720, LCM(144, 204) = 2448, LCM(204, 120) = 2040, LCM(120, 144, 204) = 12240
⇒ HCF(120, 144, 204) = [(120 × 144 × 204) × LCM(120, 144, 204)]/[LCM(120, 144) × LCM (144, 204) × LCM(204, 120)]
⇒ HCF(120, 144, 204) = (3525120 × 12240)/(720 × 2448 × 2040)
⇒ HCF(120, 144, 204) = 12.
Therefore, the HCF of 120, 144 and 204 is 12.
Example 3: Find the highest number that divides 120, 144, and 204 completely.

Solution:

The highest number that divides 120, 144, and 204 exactly is their highest common factor.
- Factors of 120 = 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
- Factors of 144 = 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
- Factors of 204 = 1, 2, 3, 4, 6, 12, 17, 34, 51, 68, 102, 204
The HCF of 120, 144, and 204 is 12.
∴ The highest number that divides 120, 144, and 204 is 12.

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FAQs on HCF of 120, 144 and 204

What is the HCF of 120, 144 and 204?

The HCF of 120, 144 and 204 is 12. To calculate the highest common factor of 120, 144 and 204, we need to factor each number (factors of 120 = 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120; factors of 144 = 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144; factors of 204 = 1, 2, 3, 4, 6, 12, 17, 34, 51, 68, 102, 204) and choose the highest factor that exactly divides 120, 144 and 204, i.e., 12.

How to Find the HCF of 120, 144 and 204 by Prime Factorization?

To find the HCF of 120, 144 and 204, we will find the prime factorization of given numbers, i.e. 120 = 2 × 2 × 2 × 3 × 5; 144 = 2 × 2 × 2 × 2 × 3 × 3; 204 = 2 × 2 × 3 × 17.
⇒ Since 2, 2, 3 are common terms in the prime factorization of 120, 144 and 204. Hence, HCF(120, 144, 204) = 2 × 2 × 3 = 12
☛ What is a Prime Number?

What are the Methods to Find HCF of 120, 144 and 204?

There are three commonly used methods to find the HCF of 120, 144 and 204.

By Long Division
By Euclidean Algorithm
By Prime Factorization

Which of the following is HCF of 120, 144 and 204? 12, 216, 244, 220, 215, 204

HCF of 120, 144, 204 will be the number that divides 120, 144, and 204 without leaving any remainder. The only number that satisfies the given condition is 12.

What is the Relation Between LCM and HCF of 120, 144 and 204?

The following equation can be used to express the relation between LCM (Least Common Multiple) and HCF of 120, 144 and 204, i.e. HCF(120, 144, 204) = [(120 × 144 × 204) × LCM(120, 144, 204)]/[LCM(120, 144) × LCM (144, 204) × LCM(120, 204)].
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