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HCF of 72, 126 and 168

HCF of 72, 126 and 168 is the largest possible number that divides 72, 126 and 168 exactly without any remainder. The factors of 72, 126 and 168 are (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72), (1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126) and (1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168) respectively. There are 3 commonly used methods to find the HCF of 72, 126 and 168 - prime factorization, Euclidean algorithm, and long division.

1.	HCF of 72, 126 and 168
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 72, 126 and 168?

Answer: HCF of 72, 126 and 168 is 6.

Explanation:

The HCF of three non-zero integers, x(72), y(126) and z(168), is the highest positive integer m(6) that divides x(72), y(126) and z(168) without any remainder.

Methods to Find HCF of 72, 126 and 168

The methods to find the HCF of 72, 126 and 168 are explained below.

Using Euclid's Algorithm
Long Division Method
Listing Common Factors

HCF of 72, 126 and 168 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(72, 126, 168) = HCF(HCF(72, 126), 168)

HCF(126, 72) = HCF(72, 126 mod 72) = HCF(72, 54)
HCF(72, 54) = HCF(54, 72 mod 54) = HCF(54, 18)
HCF(54, 18) = HCF(18, 54 mod 18) = HCF(18, 0)
HCF(18, 0) = 18 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Steps for HCF(18, 168)

HCF(168, 18) = HCF(18, 168 mod 18) = HCF(18, 6)
HCF(18, 6) = HCF(6, 18 mod 6) = HCF(6, 0)
HCF(6, 0) = 6 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of HCF of 72, 126 and 168 is 6.

HCF of 72, 126 and 168 by Long Division

HCF of 72, 126 and 168 can be represented as HCF of (HCF of 72, 126) and 168. HCF(72, 126, 168) can be thus calculated by first finding HCF(72, 126) using long division and thereafter using this result with 168 to perform long division again.

Step 1: Divide 126 (larger number) by 72 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (72) by the remainder (54). Repeat this process until the remainder = 0.
⇒ HCF(72, 126) = 18.
Step 3: Now to find the HCF of 18 and 168, we will perform a long division on 168 and 18.
Step 4: For remainder = 0, divisor = 6 ⇒ HCF(18, 168) = 6

Thus, HCF(72, 126, 168) = HCF(HCF(72, 126), 168) = 6.

HCF of 72, 126 and 168 by Listing Common Factors

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Factors of 126: 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126
Factors of 168: 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168

There are 4 common factors of 72, 126 and 168, that are 1, 2, 3, and 6. Therefore, the highest common factor of 72, 126 and 168 is 6.

☛ Also Check:

HCF of 72, 126 and 168 Examples

Example 1: Verify the relation between the LCM and HCF of 72, 126 and 168.

Solution:

The relation between the LCM and HCF of 72, 126 and 168 is given as, HCF(72, 126, 168) = [(72 × 126 × 168) × LCM(72, 126, 168)]/[LCM(72, 126) × LCM (126, 168) × LCM(72, 168)]
⇒ Prime factorization of 72, 126 and 168:
- 72 = 2 × 2 × 2 × 3 × 3
- 126 = 2 × 3 × 3 × 7
- 168 = 2 × 2 × 2 × 3 × 7
∴ LCM of (72, 126), (126, 168), (72, 168), and (72, 126, 168) is 504, 504, 504, and 504 respectively.
Now, LHS = HCF(72, 126, 168) = 6.
And, RHS = [(72 × 126 × 168) × LCM(72, 126, 168)]/[LCM(72, 126) × LCM (126, 168) × LCM(72, 168)] = [(1524096) × 504]/[504 × 504 × 504]
LHS = RHS = 6.
Hence verified.
Example 2: Find the highest number that divides 72, 126, and 168 completely.

Solution:

The highest number that divides 72, 126, and 168 exactly is their highest common factor.
- Factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
- Factors of 126 = 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126
- Factors of 168 = 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168
The HCF of 72, 126, and 168 is 6.
∴ The highest number that divides 72, 126, and 168 is 6.
Example 3: Calculate the HCF of 72, 126, and 168 using LCM of the given numbers.

Solution:

Prime factorization of 72, 126 and 168 is given as,
- 72 = 2 × 2 × 2 × 3 × 3
- 126 = 2 × 3 × 3 × 7
- 168 = 2 × 2 × 2 × 3 × 7
LCM(72, 126) = 504, LCM(126, 168) = 504, LCM(168, 72) = 504, LCM(72, 126, 168) = 504
⇒ HCF(72, 126, 168) = [(72 × 126 × 168) × LCM(72, 126, 168)]/[LCM(72, 126) × LCM (126, 168) × LCM(168, 72)]
⇒ HCF(72, 126, 168) = (1524096 × 504)/(504 × 504 × 504)
⇒ HCF(72, 126, 168) = 6.
Therefore, the HCF of 72, 126 and 168 is 6.

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FAQs on HCF of 72, 126 and 168

What is the HCF of 72, 126 and 168?

The HCF of 72, 126 and 168 is 6. To calculate the HCF of 72, 126 and 168, we need to factor each number (factors of 72 = 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72; factors of 126 = 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126; factors of 168 = 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168) and choose the highest factor that exactly divides 72, 126 and 168, i.e., 6.

Which of the following is HCF of 72, 126 and 168? 6, 217, 216, 181, 201

HCF of 72, 126, 168 will be the number that divides 72, 126, and 168 without leaving any remainder. The only number that satisfies the given condition is 6.

How to Find the HCF of 72, 126 and 168 by Prime Factorization?

To find the HCF of 72, 126 and 168, we will find the prime factorization of given numbers, i.e. 72 = 2 × 2 × 2 × 3 × 3; 126 = 2 × 3 × 3 × 7; 168 = 2 × 2 × 2 × 3 × 7.
⇒ Since 2, 3 are common terms in the prime factorization of 72, 126 and 168. Hence, HCF(72, 126, 168) = 2 × 3 = 6
☛ Prime Numbers

What is the Relation Between LCM and HCF of 72, 126 and 168?

The following equation can be used to express the relation between LCM and HCF of 72, 126 and 168, i.e. HCF(72, 126, 168) = [(72 × 126 × 168) × LCM(72, 126, 168)]/[LCM(72, 126) × LCM (126, 168) × LCM(72, 168)].
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What are the Methods to Find HCF of 72, 126 and 168?

There are three commonly used methods to find the HCF of 72, 126 and 168.

By Euclidean Algorithm
By Long Division
By Prime Factorization

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