LCM of 4 and 13
Understand the Problem
The question asks for the least common multiple (LCM) of the numbers 4 and 13. To solve this, we need to identify the smallest number that is a multiple of both 4 and 13.
Answer
$52$
Answer for screen readers
The least common multiple (LCM) of 4 and 13 is $52$.
Steps to Solve
- Identify the multiples of each number
First, we find some multiples of each number.
For 4: $$ 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, \ldots $$
For 13: $$ 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, \ldots $$
- Find the smallest common multiple
Next, we look through both lists of multiples to find the smallest number that appears in both lists.
The multiples for 4 do not overlap with the multiples of 13 until we reach 52:
- The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52
- The multiples of 13 are: 13, 26, 39, 52
Therefore, the smallest common multiple is 52.
- Confirm using prime factorization (optional)
To double-check, we can prime factorize both numbers:
- $4 = 2^2$
- $13 = 13^1$
The LCM can also be calculated using the formula: $$ \text{LCM}(a, b) = \frac{a \times b}{\text{GCD}(a, b)} $$
Since 4 and 13 are coprime (GCD is 1), the formula gives: $$ \text{LCM}(4, 13) = \frac{4 \times 13}{1} = 52 $$
The least common multiple (LCM) of 4 and 13 is $52$.
More Information
The least common multiple (LCM) of two numbers is the smallest number that can be evenly divided by both. In this case, 52 is significant because it is the first point at which the multiples of both numbers coincide.
Tips
- A common mistake is to confuse the LCM with the greatest common divisor (GCD). Ensure to differentiate between the two concepts.
- Sometimes, students stop at the largest number from one list or incorrectly identify the multiples. Always double-check that the number appears in both.
