least common multiple of 6 and 13
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 6 and 13. The LCM is the smallest number that is a multiple of both 6 and 13. To find it, we can list the multiples of each number or use the formula involving the greatest common divisor (GCD).
Answer
$78$
Answer for screen readers
The least common multiple of 6 and 13 is $78$.
Steps to Solve
-
Identify the numbers
We need to find the least common multiple (LCM) of the numbers 6 and 13. -
Find the multiples of each number
List some multiples of each number:
For 6:
- $6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...$
For 13:
- $13, 26, 39, 52, 65, 78, ...$
-
Find the common multiples
Identify the smallest number that appears in both lists. Since 6 and 13 do not share multiples until their product, let's skip to the next method. -
Use the formula involving GCD
The formula to calculate the LCM using GCD is:
$$ \text{LCM}(a, b) = \frac{|a \times b|}{\text{GCD}(a, b)} $$
In our case:
- $a = 6$
- $b = 13$
-
Calculate the GCD of 6 and 13
Since 6 and 13 have no common factors other than 1, their GCD is:
$$ \text{GCD}(6, 13) = 1 $$ -
Calculate the LCM using GCD
Now we can find the LCM:
$$ \text{LCM}(6, 13) = \frac{|6 \times 13|}{1} = \frac{78}{1} = 78 $$
The least common multiple of 6 and 13 is $78$.
More Information
The least common multiple (LCM) can also be viewed through the concept of prime factorization. However, since 6 and 13 share no prime factors (6 = $2 \times 3$ and 13 is a prime number), the LCM can be calculated efficiently using the GCD formula.
Tips
- Mixing up GCD and LCM. Remember that GCD finds the greatest common factor, while LCM finds the least common multiple.
- Failing to recognize that for two relatively prime numbers (like 6 and 13), the LCM is simply their product.
