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HCF of 12, 45 and 75

HCF of 12, 45 and 75 is the largest possible number that divides 12, 45 and 75 exactly without any remainder. The factors of 12, 45 and 75 are (1, 2, 3, 4, 6, 12), (1, 3, 5, 9, 15, 45) and (1, 3, 5, 15, 25, 75) respectively. There are 3 commonly used methods to find the HCF of 12, 45 and 75 - prime factorization, Euclidean algorithm, and long division.

1.	HCF of 12, 45 and 75
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 12, 45 and 75?

Answer: HCF of 12, 45 and 75 is 3.

Explanation:

The HCF of three non-zero integers, x(12), y(45) and z(75), is the highest positive integer m(3) that divides x(12), y(45) and z(75) without any remainder.

Methods to Find HCF of 12, 45 and 75

Let's look at the different methods for finding the HCF of 12, 45 and 75.

Long Division Method
Prime Factorization Method
Listing Common Factors

HCF of 12, 45 and 75 by Long Division

HCF of 12, 45 and 75 can be represented as HCF of (HCF of 12, 45) and 75. HCF(12, 45, 75) can be thus calculated by first finding HCF(12, 45) using long division and thereafter using this result with 75 to perform long division again.

Step 1: Divide 45 (larger number) by 12 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (12) by the remainder (9). Repeat this process until the remainder = 0.
⇒ HCF(12, 45) = 3.
Step 3: Now to find the HCF of 3 and 75, we will perform a long division on 75 and 3.
Step 4: For remainder = 0, divisor = 3 ⇒ HCF(3, 75) = 3

Thus, HCF(12, 45, 75) = HCF(HCF(12, 45), 75) = 3.

HCF of 12, 45 and 75 by Prime Factorization

Prime factorization of 12, 45 and 75 is (2 × 2 × 3), (3 × 3 × 5) and (3 × 5 × 5) respectively. As visible, 12, 45 and 75 have only one common prime factor i.e. 3. Hence, the HCF of 12, 45 and 75 is 3.

HCF of 12, 45 and 75 by Listing Common Factors

Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 45: 1, 3, 5, 9, 15, 45
Factors of 75: 1, 3, 5, 15, 25, 75

There are 2 common factors of 12, 45 and 75, that are 1 and 3. Therefore, the highest common factor of 12, 45 and 75 is 3.

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HCF of 12, 45 and 75 Examples

Example 1: Find the highest number that divides 12, 45, and 75 completely.

Solution:

The highest number that divides 12, 45, and 75 exactly is their highest common factor.
- Factors of 12 = 1, 2, 3, 4, 6, 12
- Factors of 45 = 1, 3, 5, 9, 15, 45
- Factors of 75 = 1, 3, 5, 15, 25, 75
The HCF of 12, 45, and 75 is 3.
∴ The highest number that divides 12, 45, and 75 is 3.
Example 2: Verify the relation between the LCM and HCF of 12, 45 and 75.

Solution:

The relation between the LCM and HCF of 12, 45 and 75 is given as, HCF(12, 45, 75) = [(12 × 45 × 75) × LCM(12, 45, 75)]/[LCM(12, 45) × LCM (45, 75) × LCM(12, 75)]
⇒ Prime factorization of 12, 45 and 75:
- 12 = 2 × 2 × 3
- 45 = 3 × 3 × 5
- 75 = 3 × 5 × 5
∴ LCM of (12, 45), (45, 75), (12, 75), and (12, 45, 75) is 180, 225, 300, and 900 respectively.
Now, LHS = HCF(12, 45, 75) = 3.
And, RHS = [(12 × 45 × 75) × LCM(12, 45, 75)]/[LCM(12, 45) × LCM (45, 75) × LCM(12, 75)] = [(40500) × 900]/[180 × 225 × 300]
LHS = RHS = 3.
Hence verified.
Example 3: Calculate the HCF of 12, 45, and 75 using LCM of the given numbers.

Solution:

Prime factorization of 12, 45 and 75 is given as,
- 12 = 2 × 2 × 3
- 45 = 3 × 3 × 5
- 75 = 3 × 5 × 5
LCM(12, 45) = 180, LCM(45, 75) = 225, LCM(75, 12) = 300, LCM(12, 45, 75) = 900
⇒ HCF(12, 45, 75) = [(12 × 45 × 75) × LCM(12, 45, 75)]/[LCM(12, 45) × LCM (45, 75) × LCM(75, 12)]
⇒ HCF(12, 45, 75) = (40500 × 900)/(180 × 225 × 300)
⇒ HCF(12, 45, 75) = 3.
Therefore, the HCF of 12, 45 and 75 is 3.

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FAQs on HCF of 12, 45 and 75

What is the HCF of 12, 45 and 75?

The HCF of 12, 45 and 75 is 3. To calculate the highest common factor (HCF) of 12, 45 and 75, we need to factor each number (factors of 12 = 1, 2, 3, 4, 6, 12; factors of 45 = 1, 3, 5, 9, 15, 45; factors of 75 = 1, 3, 5, 15, 25, 75) and choose the highest factor that exactly divides 12, 45 and 75, i.e., 3.

What is the Relation Between LCM and HCF of 12, 45 and 75?

The following equation can be used to express the relation between Least Common Multiple and HCF of 12, 45 and 75, i.e. HCF(12, 45, 75) = [(12 × 45 × 75) × LCM(12, 45, 75)]/[LCM(12, 45) × LCM (45, 75) × LCM(12, 75)].
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How to Find the HCF of 12, 45 and 75 by Prime Factorization?

To find the HCF of 12, 45 and 75, we will find the prime factorization of given numbers, i.e. 12 = 2 × 2 × 3; 45 = 3 × 3 × 5; 75 = 3 × 5 × 5.
⇒ Since 3 is the only common prime factor of 12, 45 and 75. Hence, HCF(12, 45, 75) = 3.
☛ What are Prime Numbers?

Which of the following is HCF of 12, 45 and 75? 3, 80, 100, 119, 89, 118

HCF of 12, 45, 75 will be the number that divides 12, 45, and 75 without leaving any remainder. The only number that satisfies the given condition is 3.

What are the Methods to Find HCF of 12, 45 and 75?

There are three commonly used methods to find the HCF of 12, 45 and 75.

By Listing Common Factors
By Prime Factorization
By Long Division

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