Relationship Between LCM and HCF: Definition, Formulas, Solved Examples

The terms HCF stands for the highest common factor and LCM stands for the least common multiple. The HCF is the most significant factor of the two numbers or more than two numbers, dividing the number exactly with no remainder. On the contrary, the LCM of two numbers or more than two numbers is the smallest number that is exactly divisible by the given numbers. After understanding the definition of HCF and LCM, we will focus on the relationship between HCF and LCM.

Lowest Common Factor (LCM) and Highest Common Factor (HCF): Definition

Lowest Common Factor (LCM)

Definition: LCM stands for Lowest or Least Common Multiple. In other words, the LCM of two or more numbers is the smallest positive integer divisible by all the given numbers.

Example: Consider this as an example; the LCM of \(12\) and \(15\) is \(60\).

To find the LCM of numbers, first, you need to mention the multiples of each given number.

Thus, the multiples of \(12=12, 24, 36, 48, 60, 72, 84…\) etc.,

Now, the multiples of the second number \(15=15, 30, 45, 60, 75, 90, 105,….\) etc,

Hence, \(60\) is the smallest number that is a multiple of both \(12\) and \(15\)

Highest Common Factor (HCF)

Definition: The greatest common factor (GCF or GCD or HCF) of a set of whole numbers is the largest positive integer that divides all the given numbers evenly with zero remainders. HCF stands for Highest Common Factor. Thus, HCF is also known as GCF (Greatest Common Factor) or GCD (Greatest Common Divisor).

Example: The HCF of \(12\) and \(15\) is \(3\). 

Now, the prime factorisation of \(12=2×2×3\)

Prime factorisation of \(15=3×5\)

As the number \(3\) is the only common factor for both the numbers \(12\) and \(15\), it is the largest number that divides both numbers.

Hence, \(3\) is the required answer.

LCM and HCF Formula

The LCM and HCF Formula are discussed below:

Relation Between LCM and HCF of Numbers

The relationship between HCF and LCM for two numbers is defined by the HCF and LCM formula given by:

\({\text{Product}}\,{\text{of}}\,{\text{two}}\,{\text{number}} = ({\text{HCF}}\,{\text{of}}\,{\text{the}}\,{\text{two}}\,{\text{number}}) \times ({\text{LCM}}\,{\text{of}}\,{\text{the}}\,{\text{two}}\,{\text{numbers}})\)

If in case you take the two numbers as m and n, then the HCF and LCM formula states that as below:

\(m \times n = {\text{HCF}}(m,\,n) \times {\text{LCM}}(m,\,n)\)

Using the given above formulas, you can calculate the HCF or LCM if you know one of them and the two numbers.

\({\text{HCF}}\,{\text{of}}\,{\text{the}}\,{\text{two}}\,{\text{numbers=}}\frac{{{\text{Product of two mumbers }}}}{{{\text{LCM of two numbers }}}}\)

\({\text{LCM}}\,{\text{of}}\,{\text{the}}\,{\text{two}}\,{\text{numbers=}}\frac{{{\text{Product of two mumbers }}}}{{{\text{HCF of two numbers }}}}\)

How to find HCF and LCM?

The two essential methods to calculate HCF and LCM for a given set of numbers is as follows:

1. Prime Factorization method
2. Division method

We will understand both methods one by one.

Finding HCF using Prime Factorisation Method

The prime factorisation method is also known as the factor tree method. Now, follow the given points to find out the HCF:

1. Write each of the numbers as a product of its prime factors.
2. Then you have to list the common factors for both the numbers.
3. All the common prime factors are the HCF (use the lower power of each common element).

Example: Finding the highest common factor of the numbers \(16\) and \(24\) .

You have to write each number as a product of its prime factors. \(16=2^{4} \quad 24=2^{3} \times 3\)
You know that the product of all the common prime factors is HCF.

The standard prime factor in this example is \(2^{3}\left[\because 2^{4}\right.\) can be written as \(\left.2^{3} \times 2\right]\) So, \({\text{HCF}} = {2^3} = 8\)

Finding HCF using the Long Division Method

Following are the steps to get the HCF of the given numbers with the Division method.

1. You have to write the given numbers horizontally and separate them by a comma. Then, draw a line under the two numbers.
2. Now, draw a horizontal line on the left side and then write the smallest prime number that divides the given numbers without leaving any remainder.
3. Next, divide the numbers by the lowest prime number and write the quotients separated by a comma under the horizontal line.
4. You have to repeat the second and the third step until you reach the stage where no common prime factor is available.
5. The numbers that are on the left-hand side will be the common prime factors of the given numbers. 
6. Thus, to find the HCF, multiply all the numbers that you get on the left side.

Finding LCM using the Prime Factorisation Method Example

To get the LCM, you need to use the maximum number of prime factors between the two or more given numbers.

Example: Find the LCM of \(8\) and \(20\) .
First, you need to write the prime factors of each of the given numbers.
So,
\(8=2 \times 2 \times 2\)
\(20=2 \times 2 \times 5\)

Now, you have to find the number of times the factors have appeared.
The factor \(2=3\) times
The factor \(5=1\) time

To get the LCM, you have to multiply each factor by the number of times it occurs in any of the numbers. Thus, you can see that the LCM of the numbers \(8\) and \(20=2 \times 2 \times 2 \times 5=40\).

Finding LCM using the Long Division Method Example

To get the LCM of a given set of numbers using the long division method, you use the same technique which is used to find the HCF using the division method. You will see the difference in the final step.

Instead of multiplying all the common factors present on the left-hand side, you have to multiply all numbers obtained at the end of the division.

Example: LCM of the numbers \(12\) and \(18\).

So, after performing the long division method, you get the LCM of the number \(12\) and \(18=2×3×2×3=36\)

LCM and HCF with Co-prime Numbers

\({\text{LCM}}\,{\text{of}}\,{\text{the}}\,{\text{Co}} – {\text{Prime}}\,{\text{numbers}}(m,\,n) = {\text{product}}\,{\text{of}}\,{\text{two}}\,{\text{numbers}}(m,\,n)\)

Given example below will be used to verify the relation.

Example: The numbers \(11\) and \(31\) are the two co-prime numbers. Now, verify the LCM of given co-prime numbers is equal to the product of the given numbers.
So, the factors of \(11\) and \(31\) are,

\(11=1 \times 11\)
\(31=1 \times 31\)
HCF of the number \(11\) and \(31=1\)
LCM of the number \(11\) and \(31=341\)
Product of the numbers \(11\) and \(31=11 \times 31=341\)
Hence proved, \({\text{LCM}}\,{\text{of}}\,{\text{the}}\,{\text{Co}} – {\text{Prime}}\,{\text{numbers}}(m,\,n) = {\text{product}}\,{\text{of}}\,{\text{two}}\,{\text{numbers}}(m,\,n)\).

LCM and HCF of Fractions

To get the HCF and LCM of the fractions like \({ }_{n}^{m}, \frac{p}{q}, \frac{u}{v}\), etc., you can use the below-given formulas:

1. \({\text{LCM}}\,{\text{of}}\,{\text{fractions}} = {\text{LCM}}\,{\text{of}}\,{\text{the}}\,{\text{Numerator}} \div {\text{HCF}}\,{\text{of}}\,{\text{the}}\,{\text{Denominators}}\)
2. \({\text{HCF}}\,{\text{of}}\,{\text{fractions}} = {\text{HCF}}\,{\text{of}}\,{\text{the}}\,{\text{Numerator}} \div {\text{LCM}}\,{\text{of}}\,{\text{the}}\,{\text{Denominators}}\)

Example: Finding out the LCM of the given fractions \(\frac{1}{4}, \frac{3}{10}, \frac{2}{5}\).

You have to use the given formula and get the HCF and LCM.

\({\text{LCM}}\,{\text{of}}\,{\text{fractions}} = {\text{LCM}}\,{\text{of}}\,{\text{the}}\,{\text{Numerators}} \div {\text{HCF}}\,{\text{of}}\,{\text{the}}\,{\text{Denominators}}\)

\({\text{LCM}}\,{\text{of}}\,{\text{fractions}} = {\text{LCM}}(1,\,3,\,2) \div {\text{HCF}}(4,\,10,\,5)\)

\(=6 \div 1=6\)

Tricks of LCM and HCF

1. The highest common factor (HCF) of two or more numbers is smaller than or equal to the smallest number of given numbers.
2. The smallest number, which is divisible by \(a, b\) and \(c\), is the lowest common multiple (LCM) of \(a, b, c\)
3. The LCM of two or more numbers is greater than or equal to the most significant number of given numbers.
4. The smallest number, when divided by \(a, b\) and \(c\) leave a remainder \(R\) in every case. Required number \( = ({\text{LCM}}\,{\text{of}}\,a,\,b,\,c) + R\).
5. The most significant number which divides \(a, b\) and \(c\) to leave the remainder \(R\) is HCF of \((a-R),(b-R)\) and \((c-R)\).
6. The most significant number which divides \(x, y, z\) to leave remainders \(a, b, c\) is \(\mathrm{HCF}\) of \((x-a),(y-b)\) and \((z-c)\).
7. The smallest number when divided by \(x, y\) and \(z\) leaves the remainder of \(a, b, c(x-a),(y-b),(z-c)\) are multiples of \(M\) Required number \( = (\operatorname{LCM} \,of\,x,y\,{\text{and}}\,z) – M\)

Solved Examples on Relationship between LCM and HCF

Q.1. Identify and write the HCF of the fractions \(\frac{4}{5}, \frac{5}{2}, \frac{6}{7}\).
Ans: Given, \(\frac{4}{5}, \frac{5}{2}, \frac{6}{7}\).
To find the HCF of the given fractions, so you have to use the below formula.
\({\text{HCF}}\,{\text{of}}\,{\text{fractions}} = {\text{HCF}}\,{\text{of}}\,{\text{Numerators}} \div {\text{LCM}}\,{\text{of}}\,{\text{Denominators}}\)
Now, \({\text{HCF}}\,{\text{of}}\,{\text{fractions}} = {\text{HCF}}(4,\,5,\,6) \div {\text{LCM}}(5,\,2,\,7)\)
Hence, \(=\frac{1}{70}\)

Q.2. The highest common factor and lowest common multiple of the two numbers are \(18\) and \(1782\), respectively. One number is \(162\) . Find the other.
Ans: Given to find the other number,
So you have to use the formula that is given below:
\({\text{HCF}} \times {\text{LCM}} = {\text{First}}\,{\text{number}} \times {\text{Second}}\,{\text{number}}\)
We get,
\(18 \times 1782 = 162 \times {\text{Second}}\,{\text{number}}\)
\(18 \times \frac{{1782}}{{162}} = {\text{Second}}\,{\text{number}}\)
Hence, the second number \(=19\).

Q.3. If the HCF of two numbers is \(3\), and their LCM is \(54\). The first number is \(27\). Find the other number.
Ans: Given to find the second number,
Use the formula that is given below:
\({\text{HCF}} \times {\text{LCM}} = {\text{Product}}\,{\text{of}}\,{\text{two}}\,{\text{numbers}}\)
\(3 \times 54 = 27 \times {\text{second}}\,{\text{number}}\)
Second number \(=\frac{3 \times 54}{27}=\frac{162}{27}=6\)
Hence, the second number \(=6\).

Q.4. When divided by \(5,7,9\) and \(12\), find the least number that leaves the same remainder \(3\) in each case.
Ans: We know that we have to find out the LCM of the divisors and then add the standard remainder with the number \(3\).
So, the \(L C M(5,7,9,12)=1260\)
Now, add the number \(3\) to \(1260 \rightarrow 3+1260=1263\)
Hence, the required answer is \(1263\).

Q.5. The two numbers ratio is \(5: 11\). HCF is 7 , so find the numbers.
Ans: We will take numbers as \(5 \mathrm{~m}\) and \(11 \mathrm{~m}\).
Since \(5: 11\) is already the reduced ratio.
So, \(m\) has to be the HCF.
Here, the numbers we get is \(5 \times 7=35\) and \(11 \times 7=77\)
Hence, the numbers are \(35\) and \(77\).

Summary

In the given article, we have covered HCF and LCM definitions and have discussed their formulas. So, you now have an understanding of how to find the LCM and the HCF of numbers using different methods like the prime factorisation method and division method. The relationship between HCF and LCM was explained along with the solved examples and a few FAQs.

FAQs on Relationship between LCM and HCF

Q.1. What is the product of HCF and LCM equal to?
Ans: The product of HCF and LCM is equal to the product of the two numbers.

Q.2. What is the relation between the LCM, HCF of the three numbers and the HCF taken by pairs?
Ans: The given formulas will help in understanding the relationship between the LCM HCF of \(3\) numbers.
\({\text{LCM}}(p,q,r) = \frac{{(p \times q \times r) \times {\text{HCF}}(p,q,r)}}{{{\text{HCF}}(p,q) \times {\text{HCF}}(q,r) \times {\text{HCF}}(r,p)}}\)
\({\text{HCF}}(p,q,r) = \frac{{(p \times q \times r) \times {\text{LCM}}(p,q,r)}}{{{\text{LCM}}(p,q) \times {\text{LCM}}(q,r) \times {\text{LCM}}(r,p)}}\)

Q.3. How do you find HCF when given LCM?
Ans:You can find out the HCF by using the given formula:
\({\text{Second}}\,{\text{number}}\,{\text{HCF}}\,{\text{of}}\,{\text{the}}\,{\text{two}}\,{\text{numbers}} = \frac{{{\text{ Product of two mumbers }}}}{{{\text{ LCM of two mumbers }}}}\)

Q.4. How is LCM used in real life?
Ans: An example is given below where LCM is used in actual life situation:
Rahul exercises every \(12\) days and Sammy every \(8\) days. Rahul and Sammy both exercised today. Calculate the days it will be until they exercise together again?
Note: This problem is solved using LCM as we are trying to figure out when the soonest (least) time will be the event of exercising continues (multiple), it will occur at the same time (Common).
Answer: LCM of the numbers \(12\) and \(8\) is \(24\) .
Hence, both of them will exercise together again in \(24\) days.

Q.5. Write the complete form of HCF and LCM?
Ans: The term HCF means Highest Common Factor, and the term LCM means Lowest Common Multiple.

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