What is the LCM of 4 and 15?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 4 and 15. To find the LCM, we look for the smallest positive integer that is a multiple of both numbers. This can be achieved by finding the prime factors of both numbers and using them to determine the LCM.
Answer
$60$
Answer for screen readers
The least common multiple (LCM) of 4 and 15 is $60$.
Steps to Solve
- Find the prime factors of each number
To find the prime factors, we divide each number by the smallest prime numbers until we can no longer divide.
For 4: $$ 4 = 2 \times 2 = 2^2 $$
For 15: $$ 15 = 3 \times 5 $$
- List all unique prime factors
From the prime factorization, we identify all unique prime factors:
- For 4, the prime factor is 2.
- For 15, the prime factors are 3 and 5.
Thus, the unique prime factors are: 2, 3, and 5.
- Determine the highest power of each prime factor
Next, we take the highest power of each prime factor from the factorizations:
- The highest power of 2 is $2^2$.
- The highest power of 3 is $3^1$.
- The highest power of 5 is $5^1$.
- Multiply the highest powers together
Now, we multiply these highest powers to find the LCM: $$ LCM = 2^2 \times 3^1 \times 5^1 $$
Calculating this gives: $$ LCM = 4 \times 3 \times 5 $$
- Calculate the final result
Now, we perform the multiplication: $$ 4 \times 3 = 12 $$ $$ 12 \times 5 = 60 $$
Therefore, the least common multiple (LCM) of 4 and 15 is 60.
The least common multiple (LCM) of 4 and 15 is $60$.
More Information
The least common multiple is frequently used in problems involving finding common denominators for fractions, synchronizing events, or scheduling tasks. The LCM of two numbers can also be computed using the greatest common divisor (GCD) through the relationship: $$ LCM(a, b) = \frac{|a \cdot b|}{GCD(a, b)}. $$
Tips
- A common mistake is to think that the LCM is simply the product of the two numbers, which can happen if the numbers have no common factors, but that's not always the case.
- Confusing LCM with GCD (greatest common divisor) is another frequent error. It's essential to remember they serve different purposes.
