Least Common Multiple (LCM) & Highest Common Factor(H.C.F)

Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the numbers.
Example: LCM of 3 and 4 = 12 because 12 is the smallest multiple which is common to 3 and 4 (In other words, 12 is the smallest number which is divisible by both 3 and 4)

LCM Example and Shortcuts

How to find out LCM using prime factorization method

We can find LCM using prime factorization method in the following steps

Step1 : Express each number as a product of prime factors.
Step2 : LCM = The product of highest powers of all prime factors

Example 1 : Find out LCM of 8 and 14
Step1 : Express each number as a product of prime factors. (Reference: Prime Factorization and how to find out Prime Factorization)
$8 = 2^{3}$
14 = 2 χ 7

Step2 : LCM = The product of highest powers of all prime factors
Here the prime factors are 2 and 7
The highest power of 2 here = $2^{3}$
The highest power of 7 here = 7
Hence LCM = $2^{3}$ χ 7

Example 2 : Find out LCM of 18, 24, 9, 36 and 90

Step1 : Express each number as a product of prime factors (Reference: Prime Factorization and how to find out Prime Factorization).

18 = 2 χ $3^{2}$
24 = $2^{3}$ χ 3
9 = $3^{2}$
36 = $2^{3}$ χ $3^{2}$
90 = 2 χ 5 χ $3^{2}$

Step2 : LCM = The product of highest powers of all prime factors
Here the prime factors are 2, 3 and 5
The highest power of 2 here = $2^{3}$
The highest power of 3 here = $3^{2}$
The highest power of 5 here = 5
Hence LCM = $2^{3}$ χ $3^{2}$ χ 5 = 360
Hence Least common multiple (L.C.M) of 18, 24, 9, 36 and 90 = 2 × 2 × 3 × 3 × 2 × 5 = 360

How to find out LCM using Division Method (shortcut)

Step 1 : Write the given numbers in a horizontal line separated by commas.
Step 2 : Divide the given numbers by the smallest prime number which can exactly divide at least two of the given numbers.
Step 3 : Write the quotients and undivided numbers in a line below the first.
Step 4 : Repeat the process until we reach a stage where no prime factor is common to any two numbers in the row.
Step 5 : LCM = The product of all the divisors and the numbers in the last line.

Example 1 : Find out LCM of 8 and 14
$2\:\begin{array}{|c}8,14\\\hline \end{array}$
$\:\begin{array}{c}&4,7\end{array}$

Example 2 : Find out LCM of 18, 24, 9, 36 and 90
$2\:\begin{array}{|c}18, 24, 9, 36, 90 \\\hline \end{array}$
$2\:\begin{array}{|c}9, 12, 9, 18, 45 \\\hline \end{array}$
$3\:\begin{array}{|c}9, 6, 9, 9, 45 \\\hline \end{array}$
$3\:\begin{array}{|c}3, 2, 3, 3, 15 \\\hline \end{array}$
$\:\begin{array}{c}&1,2,1,1,5\end{array}$
Hence Least common multiple (L.C.M) of 8 and 14 = 2 × 4 × 7 = 56

HCF Example and Shortcuts Tricks

Highest Common Factor (H.C.F) or Greatest Common Measure(G.C.M) or Greatest Common Divisor (G.C.D) of two or more numbers is the greatest number which divides each of them exactly.

Example : HCF or GCM or GCD of 60 and 75 = 15 because 15 is the highest number which divides both 60 and 75 exactly.

How to find out HCF using prime factorization method

Step1 : Express each number as a product of prime factors. (Reference: Prime Factorization and how to find out Prime Factorization)
Step2 : HCF is the product of all common prime factors using the least power of each common prime factor.

Example 1 : Find out HCF of 60 and 75 (Reference: Prime Factorization and how to find out Prime Factorization)

Step1: Express each number as a product of prime factors.
$60 = 2^{2}\times 3 \times 5$
$75 = 3\times 5^{2}$

Step2: HCF is the product of all common prime factors using the least power of each common prime factor.
Here, common prime factors are 3 and 5
The least power of 3 here = 3
The least power of 5 here = 5
Hence, HCF = 3 × 5 = 15

Example 2 : Find out HCF of 36, 24 and 12
Step1 : Express each number as a product of prime factors. (Reference: Prime Factorization and how to find out Prime Factorization)
$36 = 2^{2}\times 3^{2}$
$24 = 2^{2}\times 3$
$12 = 2^{2}\times 3$

Step2 : HCF is the product of all common prime factors using the least power of each common prime factor.
Here 2 and 3 are common prime factors.
The least power of 2 here = $2^{2}$
The least power of 3 here = 3
Hence, HCF = $2^{2}\times 3 = 12$

Example 3 : Find out HCF of 36, 27 and 80
Step1 : Express each number as a product of prime factors. (Reference: Prime Factorization and how to find out Prime Factorization)
$36 = 2^{2}\times 3^{2}$
$27 = 3^{3}$
$80 = 2^{4}\times 5$

Step2 : HCF = HCF is the product of all common prime factors using the least power of each common prime factor.
Here you can see that there are no common prime factors.
Hence, HCF = 1

How to find out HCF using prime factorization method - By dividing the numbers (shortcut)

Step 1 : Write the given numbers in a horizontal line separated by commas.
Step 2 : Divide the given numbers by the smallest prime number which can exactly divide all of the given numbers.
Step 3 : Write the quotients in a line below the first.
Step 4 : Repeat the process until we reach a stage where no common prime factor exists for all of the numbers.
Step 5 :We can see that the factors mentioned in the left side clearly divides all the numbers exactly and they are common prime factors. Their product is the HCF

Example 1 : Find out HCF of 60 and 75
$3\:\begin{array}{|c}60, 75 \\\hline \end{array}$
$5\:\begin{array}{|c}20, 25 \\\hline \end{array}$
$\:\begin{array}{c}&4, 5\end{array}$

We can see that the prime factors mentioned in the left side clearly divides all the numbers exactly and they are common prime factors. no common prime factor is exists for the numbers came at the bottom.
Hence HCF = 3 χ 5 = 15

Example 2 : Find out HCF of 36, 24 and 12
$2\:\begin{array}{|c}36, 24, 12 \\\hline \end{array}$
$2\:\begin{array}{|c}18, 12, 6 \\\hline \end{array}$
$3\:\begin{array}{|c}9, 6, 3 \\\hline \end{array}$
$\:\begin{array}{c}&3, 2, 1\end{array}$

We can see that the prime factors mentioned in the left side clearly divides all the numbers exactly and they are common prime factors. no common prime factor is exists for the numbers came at the bottom.
Hence HCF = 2 × 2 × 3 = 12.

How to calculate LCM and HCF for fractions

$Least Common Multiple (L.C.M.) for fractions = \frac{LCM of Numerators}{HCF of Denominators}$

Example 1: Find out LCM of $\frac{1}{2}, \frac{3}{8}, \frac{3}{4}$
LCM = $\frac{LCM(1, 3, 3)}{HCF(2, 8, 4)} = \frac{3}{2}$

Example 2: Find out LCM of $\frac{2}{5}, \frac{3}{10}$
LCM = $\frac{LCM(2, 3)}{HCF(5, 10)}$ = $\frac{6}{5}$

Highest Common Multiple (H.C.F) for fractions

HCF for fractions = $\frac{HCF of Numerators}{LCM of Denominators}$

Example 1: Find out HCF of $\frac{3}{5},\frac{6}{11}, \frac{9}{20}$
HCF = $\frac{HCF(3, 6, 9)}{HCF(5, 11, 20)}$ = $\frac{3}{220}$

Example 2: Find out HCF of $\frac{4}{5}$, $\frac{2}{3}$
HCF = $\frac{HCF(4, 2)}{HCF(5, 10)}$ = $\frac{2}{15}$

How to calculate LCM and HCF for Decimals

Step 1 : Make the same number of decimal places in all the given numbers by suffixing zero(s) in required numbers as needed.
Step 2 : Now find the LCM/HCF of these numbers without decimal.
Step 3 : Put the decimal point in the result obtained in step 2 leaving as many digits on its right as there are in each of the numbers.

Example1 : Find the LCM and HCF of .63, 1.05, 2.1
Step 1 : Make the same number of decimal places in all the given numbers by suffixing zero(s) in required numbers as needed.
i.e., the numbers can be writtten as .63, 1.05, 2.10

Step 2 : Now find the LCM/HCF of these numbers without decimal.
Without decimal, the numbers can be written as 63, 105 and 210 .
LCM (63, 105 and 210) = 630
HCF (63, 105 and 210) = 21

Step 3 : Put the decimal point in the result obtained in step 2 leaving as many digits on its right as there are in each of the numbers.
i.e., here, we need to put decimal point in the result obtained in step 2 leaving two digits on its right.
i.e., the LCM (.63, 1.05, 2.1) = 6.30
HCF (.63, 1.05, 2.1) = .21

LCM and HCF Problems from Previous Year Questions

LCM and HCF Problems from Previous Year Questions