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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

  1. Identify the numbers
    We need to find the least common multiple (LCM) of the numbers 6 and 13.

  2. 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:

  • $13, 26, 39, 52, 65, 78, ...$
  1. 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.

  2. 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:

  • $a = 6$
  • $b = 13$
  1. Calculate the GCD of 6 and 13
    Since 6 and 13 have no common factors other than 1, their GCD is:
    $$ \text{GCD}(6, 13) = 1 $$

  2. 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.
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