LCM and HCF Shortcut Tricks and Formulas

LCM and HCF Shortcut Tricks - Page 2
LCM and HCF Important Questions - Page 3
LCM and HCF Video Lecture - Page 4

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