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HCF of 90 and 105

HCF of 90 and 105 is the largest possible number that divides 90 and 105 exactly without any remainder. The factors of 90 and 105 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 and 1, 3, 5, 7, 15, 21, 35, 105 respectively. There are 3 commonly used methods to find the HCF of 90 and 105 - long division, Euclidean algorithm, and prime factorization.

1.	HCF of 90 and 105
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 90 and 105?

Answer: HCF of 90 and 105 is 15.

Explanation:

The HCF of two non-zero integers, x(90) and y(105), is the highest positive integer m(15) that divides both x(90) and y(105) without any remainder.

Methods to Find HCF of 90 and 105

The methods to find the HCF of 90 and 105 are explained below.

Using Euclid's Algorithm
Long Division Method
Listing Common Factors

HCF of 90 and 105 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.

Here X = 105 and Y = 90

HCF(105, 90) = HCF(90, 105 mod 90) = HCF(90, 15)
HCF(90, 15) = HCF(15, 90 mod 15) = HCF(15, 0)
HCF(15, 0) = 15 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of HCF of 90 and 105 is 15.

HCF of 90 and 105 by Long Division

HCF of 90 and 105 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 105 (larger number) by 90 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (90) by the remainder (15).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (15) is the HCF of 90 and 105.

HCF of 90 and 105 by Listing Common Factors

Factors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Factors of 105: 1, 3, 5, 7, 15, 21, 35, 105

There are 4 common factors of 90 and 105, that are 1, 3, 5, and 15. Therefore, the highest common factor of 90 and 105 is 15.

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HCF of 90 and 105 Examples

Example 1: Find the highest number that divides 90 and 105 exactly.

Solution:

The highest number that divides 90 and 105 exactly is their highest common factor, i.e. HCF of 90 and 105.
⇒ Factors of 90 and 105:
- Factors of 90 = 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
- Factors of 105 = 1, 3, 5, 7, 15, 21, 35, 105
Therefore, the HCF of 90 and 105 is 15.
Example 2: For two numbers, HCF = 15 and LCM = 630. If one number is 105, find the other number.

Solution:

Given: HCF (z, 105) = 15 and LCM (z, 105) = 630
∵ HCF × LCM = 105 × (z)
⇒ z = (HCF × LCM)/105
⇒ z = (15 × 630)/105
⇒ z = 90
Therefore, the other number is 90.
Example 3: Find the HCF of 90 and 105, if their LCM is 630.

Solution:

∵ LCM × HCF = 90 × 105
⇒ HCF(90, 105) = (90 × 105)/630 = 15
Therefore, the highest common factor of 90 and 105 is 15.

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FAQs on HCF of 90 and 105

What is the HCF of 90 and 105?

The HCF of 90 and 105 is 15. To calculate the Highest common factor of 90 and 105, we need to factor each number (factors of 90 = 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90; factors of 105 = 1, 3, 5, 7, 15, 21, 35, 105) and choose the highest factor that exactly divides both 90 and 105, i.e., 15.

What are the Methods to Find HCF of 90 and 105?

There are three commonly used methods to find the HCF of 90 and 105.

By Prime Factorization
By Euclidean Algorithm
By Long Division

What is the Relation Between LCM and HCF of 90, 105?

The following equation can be used to express the relation between Least Common Multiple (LCM) and HCF of 90 and 105, i.e. HCF × LCM = 90 × 105.

If the HCF of 105 and 90 is 15, Find its LCM.

HCF(105, 90) × LCM(105, 90) = 105 × 90
Since the HCF of 105 and 90 = 15
⇒ 15 × LCM(105, 90) = 9450
Therefore, LCM = 630
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How to Find the HCF of 90 and 105 by Long Division Method?

To find the HCF of 90, 105 using long division method, 105 is divided by 90. The corresponding divisor (15) when remainder equals 0 is taken as HCF.

How to Find the HCF of 90 and 105 by Prime Factorization?

To find the HCF of 90 and 105, we will find the prime factorization of the given numbers, i.e. 90 = 2 × 3 × 3 × 5; 105 = 3 × 5 × 7.
⇒ Since 3, 5 are common terms in the prime factorization of 90 and 105. Hence, HCF(90, 105) = 3 × 5 = 15
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