LCM (26; 16) = ? Calculate the least common multiple, LCM, by two methods: 1) The prime factorization of the numbers and 2) The Euclidean algorithm

lcm (26; 16) = ?

Method 1. The prime factorization:

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


26 = 2 × 13
26 is not a prime number but a composite one.


16 = 24
16 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 (26; 16) = 24 × 13



lcm (26; 16) = 24 × 13 = 208
The two numbers have common prime factors

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:
26 ÷ 16 = 1 + 10
Step 2. Divide the smaller number by the above operation's remainder:
16 ÷ 10 = 1 + 6
Step 3. Divide the remainder of the step 1 by the remainder of the step 2:
10 ÷ 6 = 1 + 4
Step 4. Divide the remainder of the step 2 by the remainder of the step 3:
6 ÷ 4 = 1 + 2
Step 5. Divide the remainder of the step 3 by the remainder of the step 4:
4 ÷ 2 = 2 + 0
At this step, the remainder is zero, so we stop:
2 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 (26; 16) = 2


Calculate the least common multiple:

The least common multiple, Formula:

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


lcm (26; 16) =


(26 × 16) / gcf, hcf, gcd (26; 16) =


416 / 2 =


208


lcm (26; 16) = 208 = 24 × 13

The final answer:
The least common multiple
lcm (26; 16) = 208 = 24 × 13
The two numbers have common prime factors.

Why is it useful to calculate the least common multiple?

When adding, subtracting or sorting fractions with different denominators, in order to work with those fractions we must first make the denominators the same. An easy way is to calculate the least common multiple of all the denominators of the fractions (the least common 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.


The least common multiple, LCM: the latest 5 calculated values

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). 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