LCM (165; 2,012) = ? Calculate the least common multiple, LCM, by two methods: 1) The prime factorization of the numbers and 2) The Euclidean algorithm

lcm (165; 2,012) = ?

Method 1. The prime factorization:

The prime factorization of a number: finding the prime numbers that multiply together to make that number.


165 = 3 × 5 × 11
165 is not a prime number but a composite one.


2,012 = 22 × 503
2,012 is not a prime number but a composite one.


* The natural numbers that are only divisible by 1 and themselves are called prime numbers. A prime number has exactly two factors: 1 and itself.
* A composite number is a natural number that has at least one other factor than 1 and itself.



Calculate the least common multiple, lcm:

Multiply all the prime factors of the two numbers, taken by the largest exponents (largest powers).


lcm (165; 2,012) = 22 × 3 × 5 × 11 × 503



lcm (165; 2,012) = 22 × 3 × 5 × 11 × 503 = 331,980
The two numbers have no prime factors in common:
331,980 = 165 × 2,012

Method 2. The Euclidean Algorithm:

Calculate the greatest (highest) common factor (divisor):

This algorithm involves the process of dividing numbers and calculating the remainders.


'a' and 'b' are the two natural numbers, 'a' >= 'b'.


Divide 'a' by 'b' and get the remainder of the operation, 'r'.


If 'r' = 0, STOP. 'b' = the gcf (hcf, gcd) of 'a' and 'b'.


Else: Replace ('a' by 'b') and ('b' by 'r'). Return to the step above.



Step 1. Divide the larger number by the smaller one:
2,012 ÷ 165 = 12 + 32
Step 2. Divide the smaller number by the above operation's remainder:
165 ÷ 32 = 5 + 5
Step 3. Divide the remainder of the step 1 by the remainder of the step 2:
32 ÷ 5 = 6 + 2
Step 4. Divide the remainder of the step 2 by the remainder of the step 3:
5 ÷ 2 = 2 + 1
Step 5. Divide the remainder of the step 3 by the remainder of the step 4:
2 ÷ 1 = 2 + 0
At this step, the remainder is zero, so we stop:
1 is the number we were looking for - the last non-zero remainder.
This is the greatest (highest) common factor (divisor).


The greatest (highest) common factor (divisor):
gcf, hcf, gcd (165; 2,012) = 1


Calculate the least common multiple:

The least common multiple, Formula:

lcm (a; b) = (a × b) / gcf, hcf, gcd (a; b)


lcm (165; 2,012) =


(165 × 2,012) / gcf, hcf, gcd (165; 2,012) =


331,980 / 1 =


331,980


lcm (165; 2,012) = 331,980 = 22 × 3 × 5 × 11 × 503

The final answer:
The least common multiple
lcm (165; 2,012) = 331,980 = 22 × 3 × 5 × 11 × 503
The two numbers have no prime factors in common:
331,980 = 165 × 2,012

Why do we need to calculate the greatest common factor?

Once you've calculated the greatest common factor of the numerator and the denominator of a fraction, it becomes much easier to fully reduce (simplify) the fraction to the lowest terms (the smallest possible numerator and denominator).


By definition, the least common multiple of two numbers is the smallest natural number that is: (1) greater than 0 and (2) a multiple of both numbers.


Other operations of this type:


Calculator: calculate the least common multiple, lcm

Calculate the least common multiple of the numbers, LCM:

Method 1: Run the prime factorization of the numbers - then multiply all the prime factors of the numbers, taken by the largest exponents.

Method 2: The Euclidean algorithm:
lcm (a; b) = (a × b) / gcf (a; b) .

Method 3: The divisibility of the numbers.

The least common multiple, LCM: the latest calculated

The LCM of 165 and 2,012 = ? May 16 09:07 UTC (GMT)
The LCM of 360 and 12 = ? May 16 09:07 UTC (GMT)
The LCM of 341,433 and 2,390,080 = ? May 16 09:07 UTC (GMT)
The LCM of 36 and 885 = ? May 16 09:07 UTC (GMT)
The LCM of 150 and 6,038 = ? May 16 09:07 UTC (GMT)
The LCM of 484 and 225 = ? May 16 09:07 UTC (GMT)
The LCM of 119 and 102 = ? May 16 09:07 UTC (GMT)
The LCM of 4,600 and 460 = ? May 16 09:07 UTC (GMT)
The LCM of 59,049 and 1,310,720 = ? May 16 09:06 UTC (GMT)
The LCM of 14,280 and 4,199 = ? May 16 09:06 UTC (GMT)
The LCM of 4,046,132 and 4,050,156 = ? May 16 09:06 UTC (GMT)
The LCM of 5,711 and 8 = ? May 16 09:06 UTC (GMT)
The LCM of 31,690 and 253,584 = ? May 16 09:06 UTC (GMT)
The least common multiple, LCM: the list of all the operations

The least common multiple (lcm). What it is and how to calculate it.


What is a prime number? Definition, examples

What is a composite number? Definition, examples

The prime numbers up to 1,000

The prime numbers up to 10,000

The Sieve of Eratosthenes

The Euclidean Algorithm

Completely reduce (simplify) fractions to the lowest terms: Steps and Examples