Least Common Multiple (LCM)
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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 number which is a multiple of both 3 and 4 (In other words, 12 is the smallest number which is divisible by both 3 and 4).
We can find LCM using prime factorization method or division method.
Example: LCM of 3 and 4 = 12 because 12 is the smallest number which is a multiple of both 3 and 4 (In other words, 12 is the smallest number which is divisible by both 3 and 4).
We can find LCM using prime factorization method or division method.
How to find out LCM using prime factorization methodStep 1 : Express each number as a product of prime factors.
Step 2 : LCM = The product of highest powers of all prime factors.
Example 1: Find out LCM of 8 and 14Step 1 : Express each number as a product of prime factors. (Reference: Prime Factorization)
8 = 23
14 = 2 × 7
Step 2 : 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 = 56Example 2: Find out LCM of 18, 24, 9, 36 and 90
Step 1 : Express each number as a product of prime factors.
18 = 2 × 32
24 = 23 × 3
9 = 32
36 = 23 × 32
90 = 2 × 5 × 32
Step 2 : 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
Step 2 : LCM = The product of highest powers of all prime factors.
Example 1: Find out LCM of 8 and 14Step 1 : Express each number as a product of prime factors. (Reference: Prime Factorization)
8 = 23
14 = 2 × 7
Step 2 : 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 = 56Example 2: Find out LCM of 18, 24, 9, 36 and 90
Step 1 : Express each number as a product of prime factors.
18 = 2 × 32
24 = 23 × 3
9 = 32
36 = 23 × 32
90 = 2 × 5 × 32
Step 2 : 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 method)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
= 56Example 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
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 | |
| 4,7 |
= 2 × 4 × 7
= 56Example 2: Find out LCM of 18, 24, 9, 36 and 90
| 2 | |
| 2 | |
| 3 | |
| 3 | |
| 1,2,1,1,5 |
= 2 × 2 × 3 × 3 × 2 × 5
= 360
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