What is Least Common Multiple (LCM) and how to find LCM
ব্যাংক জব নিয়োগ পরীক্ষা গণিত LCM HCF
Least Common Multiple (LCM)
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)
We can find out LCM using prime factorization method or division method
Step1 : Express each number as a product of prime factors.
Step2 : LCM = The product of highest powers of all prime factors
How to find out LCM using prime factorization method
Example 1 : Find out LCM of 8 and 14
Step1 : Express each number as a product of prime factors.
8 = 23
14 = 2 × 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 = 23
The highest power of 7 here = 7
Hence LCM = 23 × 7 = 56
Example 2 : Find out LCM of 18, 24, 9, 36 and 90
Step1 : Express each number as a product of prime factors
18 = 2 × 32
24 = 23 × 3
9 = 32
36 = 23 × 32
90 = 2 × 5 × 32
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 = 23
The highest power of 3 here = 32
The highest power of 5 here = 5
Hence LCM = 23 × 32 × 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
Hence Least common multiple (L.C.M) of 8 and 14 = 2 × 4 × 7 = 56
Example 2 : Find out LCM of 18, 24, 9, 36 and 90
Hence Least common multiple (L.C.M) of 18, 24, 9, 36 and 90 = 2 × 2 × 3 × 3 × 2 × 5 = 360
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.
We can find out HCF using prime factorization method or division 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
Step1 : Express each number as a product of prime factors.
60 = 22 × 3 × 5
75 = 3 × 52
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.
36 = 22 × 32
24 = 23 × 3
12 = 22 × 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 = 22
The least power of 3 here = 3
Hence, HCF = 22 × 3 = 12
Example 3 : Find out HCF of 36, 27 and 80
Step1 : Express each number as a product of prime factors.
36 = 22 × 32
27 = 33
80 = 24 × 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
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
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