What is the difference between the LCM and the HCF of the numbers 20 30 and 40?

The highest common factor and least common multiple are such concepts that are important not only for the school level but also for several exams.These exams might be MAT, CAT, and different recruitment exams.So, it is important to understand the basic concepts and properties of these two properly.

In this article, we’re going to discuss what is LCM and HCF, their properties & differences, and how to calculate these properly with suitable examples.

What are LCM and HCF?

LCM is the smallest whole number that occurs in both of their time tables and it is the smallest integer that is a multiple of both numbers.It is also known as the lowest common multiple and shows the least number divisible by 2 or more numbers.For example, suppose two numbers 4 and 5.The common multiples of 4 are:4, 8, 12, 16, 20, 24, 28And of 5 are:5, 10, 15, 20, 25.The LCM of two different numbers 4 & 5 is 20.

Whereas, HCF is the highest common factor between different numbers.

The HCF of these two numbers is 6 because it is the biggest number that divides each of these numbers.

Difference between HCF and LCM

LCM is the least common multiple and HCF is the highest common factor between different numbers.The main purpose of showing the difference between these two numbers is to show the difference between a factor and a multiple.

A multiple of a whole number is an integer that occurs in the timetable.

Whereas, the factor of an integer is that number that divides the integer with no reminder at the end.

And the exact LCM of 15 and 18 is 90.

Thus, the product of L.C.M and H.C.F is equal to the product of these two numbers.

Properties of LCM and HCF

Some of the main properties of these two concepts are:

• The highest common factor of two or more than two prime numbers is always 1.
• The LCM of two or more than two numbers will be their products.
• The HCF of the given numbers is never greater than the given numbers.
• The least common multiple of the given numbers is never smaller than any of the given numbers
• The product of two different numbers ‘a’ and ‘b’ is always equal to the

product of their least common multiples and HCF.
Further in this article, we’ll discuss that:

How to Calculate HCF and LCM properly?

You can use the mentioned below ways to calculate the LCM or HCF of two numbers:

⦁ Prime Factorization Method for LCM and HCF

To calculate the least common factor of random two numbers 45 and 60 simply follow:

• List the prime factors of the given numbers as:45 = 2 x 2 x 3 x 5

  60 = 3 x 3 x 5

• Multiply each factor that occurs for the maximum times

LCM = 2 x 2 x 3 x 3 x 5 = 180.
And to calculate the HCF by the prime factorization method take two random numbers 104 and 144.The prime factors of these two numbers are:104 = 23 x 13144 = 2 x 2 x 2 x 2 x 3 x 3

The most common factors of these two numbers are 2 x 2 x 2.

HCF = 2 x 2 x 2 = 8.

⦁ Division Method for LCM and HCF

To calculate the least common multiple “LCM” of two different numbers 24 and 15, follow the mentioned below steps:

• Divide the given number by the lowest prime number
• Write number and quotient which are not divisible the least prime number
• Further, divide these numbers with the other prime number
• Keep the division number until the reminder becomes the prime number or 1.
• Multiply the divisors and other prime numbers to get the LCM.

And the LCM of the two numbers 24 and 15 is:

LCM of these two numbers is:
LCM = 2 x 2 x 2 x 3 x 5 = 120.
To calculate HCF by division method, simply follow the mentioned below steps:

• Divide the larger number by the smaller one
• Further, divide the divisor of the above one by the reminder
• Keep dividing the divisor of step 2 with the reminder until the remainder becomes zero.

For example, suppose two different numbers 12 and 18.
The greatest common factor of these numbers is:

The HCF of these two numbers by the division method is 6.

⦁ Using Online Calculators

Several useful online calculators are available to find the greatest common factor.These converters allow you to check the HCF, LCM, or GCD of different numbers with just a single click.These online calculators also allow you to select the desired method to find the LCM or HCF.

Most of the tools provide the following methods to check the highest common factor and least common multiple:

• Prime Factorization Method
• Division Method
• Euclidean Method
• Binary stein’s Algorithm
• And more

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HCF of 20, 30 and 40 is the largest possible number that divides 20, 30 and 40 exactly without any remainder. The factors of 20, 30 and 40 are (1, 2, 4, 5, 10, 20), (1, 2, 3, 5, 6, 10, 15, 30) and (1, 2, 4, 5, 8, 10, 20, 40) respectively. There are 3 commonly used methods to find the HCF of 20, 30 and 40 - prime factorization, long division, and Euclidean algorithm.

What is HCF of 20, 30 and 40?

Answer: HCF of 20, 30 and 40 is 10.

Explanation:

The HCF of three non-zero integers, x(20), y(30) and z(40), is the highest positive integer m(10) that divides x(20), y(30) and z(40) without any remainder.

Methods to Find HCF of 20, 30 and 40

Let's look at the different methods for finding the HCF of 20, 30 and 40.

• Listing Common Factors
• Long Division Method
• Prime Factorization Method

HCF of 20, 30 and 40 by Listing Common Factors

• Factors of 20: 1, 2, 4, 5, 10, 20
• Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
• Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40

There are 4 common factors of 20, 30 and 40, that are 1, 2, 10, and 5. Therefore, the highest common factor of 20, 30 and 40 is 10.

HCF of 20, 30 and 40 by Long Division

HCF of 20, 30 and 40 can be represented as HCF of (HCF of 20, 30) and 40. HCF(20, 30, 40) can be thus calculated by first finding HCF(20, 30) using long division and thereafter using this result with 40 to perform long division again.

• Step 1: Divide 30 (larger number) by 20 (smaller number).
• Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (20) by the remainder (10). Repeat this process until the remainder = 0.
  ⇒ HCF(20, 30) = 10.
• Step 3: Now to find the HCF of 10 and 40, we will perform a long division on 40 and 10.
• Step 4: For remainder = 0, divisor = 10 ⇒ HCF(10, 40) = 10

Thus, HCF(20, 30, 40) = HCF(HCF(20, 30), 40) = 10.

HCF of 20, 30 and 40 by Prime Factorization

Prime factorization of 20, 30 and 40 is (2 × 2 × 5), (2 × 3 × 5) and (2 × 2 × 2 × 5) respectively. As visible, 20, 30 and 40 have common prime factors. Hence, the HCF of 20, 30 and 40 is 2 × 5 = 10.

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HCF of 20, 30 and 40 Examples

1. Example 1: Calculate the HCF of 20, 30, and 40 using LCM of the given numbers.

   Solution:

   Prime factorization of 20, 30 and 40 is given as,

   • 20 = 2 × 2 × 5
   • 30 = 2 × 3 × 5
   • 40 = 2 × 2 × 2 × 5

   LCM(20, 30) = 60, LCM(30, 40) = 120, LCM(40, 20) = 40, LCM(20, 30, 40) = 120 ⇒ HCF(20, 30, 40) = [(20 × 30 × 40) × LCM(20, 30, 40)]/[LCM(20, 30) × LCM (30, 40) × LCM(40, 20)] ⇒ HCF(20, 30, 40) = (24000 × 120)/(60 × 120 × 40) ⇒ HCF(20, 30, 40) = 10.

   Therefore, the HCF of 20, 30 and 40 is 10.

Show Solution >

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The HCF of 20, 30 and 40 is 10. To calculate the highest common factor (HCF) of 20, 30 and 40, we need to factor each number (factors of 20 = 1, 2, 4, 5, 10, 20; factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30; factors of 40 = 1, 2, 4, 5, 8, 10, 20, 40) and choose the highest factor that exactly divides 20, 30 and 40, i.e., 10.

Which of the following is HCF of 20, 30 and 40? 10, 79, 51, 88, 54, 83, 62, 41

HCF of 20, 30, 40 will be the number that divides 20, 30, and 40 without leaving any remainder. The only number that satisfies the given condition is 10.

How to Find the HCF of 20, 30 and 40 by Prime Factorization?

To find the HCF of 20, 30 and 40, we will find the prime factorization of given numbers, i.e. 20 = 2 × 2 × 5; 30 = 2 × 3 × 5; 40 = 2 × 2 × 2 × 5. ⇒ Since 2, 5 are common terms in the prime factorization of 20, 30 and 40. Hence, HCF(20, 30, 40) = 2 × 5 = 10

☛ What is a Prime Number?

What are the Methods to Find HCF of 20, 30 and 40?

There are three commonly used methods to find the HCF of 20, 30 and 40.

• By Long Division
• By Prime Factorization
• By Listing Common Factors

What is the Relation Between LCM and HCF of 20, 30 and 40?

The following equation can be used to express the relation between LCM (Least Common Multiple) and HCF of 20, 30 and 40, i.e. HCF(20, 30, 40) = [(20 × 30 × 40) × LCM(20, 30, 40)]/[LCM(20, 30) × LCM (30, 40) × LCM(20, 40)].
☛ HCF Calculator