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To find the least common multiple (LCM) of \( 8a \) and \( 10a^3 \), follow these steps:

### Step 1: Factor the coefficients
- The coefficients are 8 and 10.
- \( 8 = 2^3 \) (prime factorization of 8)
- \( 10 = 2 \times 5 \) (prime factorization of 10)

### Step 2: Identify the highest powers of the prime factors
- From 8, we get \( 2^3 \).
- From 10, we get \( 2^1 \) and \( 5^1 \).

The LCM takes the highest power of each prime:
- \( 2^3 \) (from 8)
- \( 5^1 \) (from 10)

Thus, the LCM of 8 and 10 is:
\[
\text{LCM}(8, 10) = 2^3 \times 5 = 8 \times 5 = 40
\]

### Step 3: Handle the variables
For the variable \( a \), take the highest power. Between \( a \) and \( a^3 \), the highest power is \( a^3 \).

### Step 4: Combine the results
The LCM of \( 8a \) and \( 10a^3 \) is:
\[
\text{LCM}(8a, 10a^3) = 40a^3
\]

Thus, the least common multiple is \( \boxed{40a^3} \).

Would you like more details or have any questions?

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Here are some follow-up questions to enhance your understanding:
1. What is the prime factorization of 12?
2. How do you find the greatest common divisor (GCD) of two terms?
3. What is the LCM of \( 15a^2 \) and \( 20a^3 \)?
4. Can you explain the difference between GCD and LCM?
5. How would the answer change if the terms were \( 12a \) and \( 8a^2 \)?

**Tip:** When finding the LCM, always take the highest powers of all prime factors and variables involved.

Find the least common multiple (LCM) of 8a and 10a^3.

Learn how to calculate the least common multiple (LCM) of two algebraic terms, 8a and 10a^3. This problem covers key concepts in algebra, including LCM using prime factorization and variables. Suitable for students in Grades 7-9.

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