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?

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.