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HCF and LCM

HCF and LCM stand for Highest Common Factor and Lowest Common Multiple respectively. HCF, also called the Greatest Common Divisor(GCD), is defined as the most crucial factor present in between given two or more numbers.

In contrast, LCM represents the least number, exactly divisible by two or more numbers

HCF and LCM
HCF & LCM (Source)

LCM

In number theory, the least common multiple of two integers, ‘a’ and ‘b,’ are usually denoted by \(lcm(a, b)\). It can also be defined as the smallest positive integer which is divisible by both ‘a’ and ‘b’.

LCM using HCF or Greatest Common Divisor(GCD)

The formula for LCM of any two nos. = \(\frac{a*b}{gcd(a,b)}\)

LCM formula for any two fractions = \(\frac{\mbox{L.C.M. of the Numerator}}{\mbox{H.C.M. of the denominator}}\).

Example 
\(lcm(6,21) = \frac{6*21}{gcd(6,21)} = \frac{6*21}{3} = 7*6 =42\)

LCM using Prime Factorization

From the fundamental theorem of arithmetic, we know that every integer greater than 1 is either itself is a prime number or it can be expressed as the product of prime numbers.

Then factorization theorem indicates that every positive integer greater than one can only be written in one way as a product of prime numbers. So, using the method of prime factorisation we can easily find out LCM. Check out the example below to understand.

\(lcm(6,21)\); 
Prime factorisation of 6 = \(1 \cdot 2 \cdot 3\)
Prime factorisation of 21 = \(1 \cdot 3 \cdot 7\)

To find the LCM, we need to multiply all the prime factors but if there is a factor which occurs in the prime factorisation of both the numbers, we need to take it once only.

Here, there is only one common multiple, i.e. 3

Therefore, the \(lcm(6,21) = 2 \cdot 3 \cdot 7 = 42\).

Using the Table-Method   

It begins by listing all of the given numbers in a table (vertically),

(Example: 6, 14, 21, 25)
6
14
21
24

The process commences by dividing all of the numbers by 2. If 2 divides any of them evenly, then write 2 in a new column at the top of the table, and the result of division by 2 of each number is in the new column. Rewrite the number again if it is not evenly divisible.

x2
63
147
2121
2525

Now, as there are no more factors of 2, we shall move on to next prime no. i.e. 3.

x23
631
1477
21217
252525

Now to 5

x235
6311
14777
212177
25252525

As, we have one more term of 5 we will rewrite it again,

x2355
63111
147777
2121777
25252551

And finally now 7.

x23557
631111
1477771
21217771
252525511

Now, we will multiply the top row,
=> \(2 \cdot 3 \cdot 5 \cdot 5 \cdot 7 = 1050\). 

HCF

In number theory, the highest common factor or greatest common divisor(gcd) of two integers, ‘a’ and ‘b,’ is usually denoted by \(gcd(a, b)\), which is the largest positive integer that divides both ‘a’ and ‘b.’

Note we will be using \(hcf(a, b)\) notation in the article to avoid confusion

HCF using Prime Factorisation

The factorization theorem indicates that every positive integer greater than one can only be written in one way as a product of prime numbers. 

Example

\(hcf(16,24)\), 
Prime factorisation of 16 = \(1 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\)
Prime factorisation of 24 = \(1 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3\)

In contrast to LCM, we need to pick only the common factors of both the numbers and multiply them to get the HCF.

Here, there are four common multiple, i.e. 2,2,2,2.

Therefore, the \(hcf(16,24)    = 2 \cdot 2 \cdot 2 \cdot 2 = 8\)

HCF using LCM

\(hcf\) formula for any two nos. = \(\frac{a*b}{lcm(a,b)}\)

Example
\(hcf(16,24) = \frac{16*24}{lcm(16,24)} = \frac{16*24}{48} = 8\)

Applications of HCF & LCM

LCM

Operating With Fractions

When we add, subtract, or compare simple fractions, the L.C.M. of the denominators is used. For example,
\(\frac{2}{21}+\frac{1}{6}=\frac{4}{42}+\frac{7}{42}= \frac{11}{42}\)

Here, the denominator 42 was used because it is the L.C.M. of 21 and 6.

Gears Problem

Suppose there are two meshing gears in a machine, having ‘x’ and ‘y’ teeth, respectively, and the gears are marked by a line segment drawn from the center of the first gear to the center of the second gear. When the gears begin rotating, the number of rotations the first gear must complete realigning the line segment can be calculated using \(lcm( m, n)\).

HCF

Reducing Fractions

The HCF is useful for reducing fractions to the lowest terms. For example, \(hcf(42, 56) = 14\),

Therefore,
\(\frac {42}{56} = \frac {3\cdot 14}{4\cdot 14} = \frac {3}{4}\).

Finding Least Common Multiple

The least common multiple can be found out by the greatest common divisor of two numbers, using the relation,
\(lcm( a , b ) = \frac {|a \cdot b|}{ hcf( a , b )}\).

Finding Coprimes

If HCF of two numbers is 1 then they are called Coprime numbers.

Questions on HCF & LCM

Question 1. What is the greatest number that will divide 18, 30 & 11 so as to leave the same remainder in each case? 

Solution. Prime factors of;
\(18 = 2 \cdot 3 \cdot 3\)
\(30 = 2 \cdot 3 \cdot 5\)
\(11 = 1 \cdot 1\)

\(hcf(18, 30, 11) = 2 \cdot 3 \cdot 3 \cdot 5 \cdot 11 = 990\)

Question 2. Three number are in the ratio of 3 : 4 : 5 and their L.C.M. is 2400. Their H.C.F. is:

Solution. Let the three nos. be 3x, 4x and 5x
LCM = 60x
60x = 2400
=> x = 40

=> the nos. are  (3 x 40), (4 x 40) & (5 x 40)

Therefore the required H.C.F. = 40

FAQs

What is meant by HCF?

HCF stands for ‘highest common factor’. HCF is also known as Greatest Common Divisor (GCD). It is the greatest number that can divide the given numbers.

What is the HCF of two consecutive numbers?

The HCF of two consecutive numbers is always one.

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