lcm of 15 and 16
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 15 and 16. To solve this, we need to determine the smallest positive integer that is divisible by both 15 and 16.
Answer
$240$
Answer for screen readers
The least common multiple (LCM) of 15 and 16 is $240$.
Steps to Solve
- Factor the Numbers
First, we find the prime factorization of both numbers.
For 15: $$ 15 = 3 \times 5 $$
For 16: $$ 16 = 2^4 $$
- Identify the Highest Powers
Next, we identify the highest power of each prime factor that appears in the factorizations.
From the factorizations:
- The prime factor 2 appears in 16 as $2^4$.
- The prime factor 3 appears in 15 as $3^1$.
- The prime factor 5 appears in 15 as $5^1$.
- Multiply the Highest Powers
Now we multiply the highest powers of all prime factors together to find the LCM.
The LCM is given by: $$ LCM = 2^4 \times 3^1 \times 5^1 $$
Calculating this step-by-step:
- First calculate $2^4 = 16$.
- Then calculate $3^1 = 3$.
- Finally calculate $5^1 = 5$.
Thus, we compute: $$ LCM = 16 \times 3 \times 5 $$
- Perform the Multiplication
Now, let's perform the multiplication:
- Calculate $16 \times 3 = 48$.
- Then multiply $48 \times 5$.
Finally, $$ 48 \times 5 = 240 $$
Therefore, the least common multiple of 15 and 16 is 240.
The least common multiple (LCM) of 15 and 16 is $240$.
More Information
The least common multiple is useful in various applications, including finding a common denominator in fractions, scheduling events that occur at different intervals, and problems in number theory.
Tips
- Forgetting to factor completely, which can lead to incorrect highest powers.
- Incorrectly multiplying prime factors, especially if multiple calculations are involved; it's best to tackle multiplications step-by-step.
