Find the LCM for   30 and 36 showing the factor tree in the answer

To find the Least Common Multiple (LCM) of 30 and 36, let's follow the steps similar to before.

### Step 1: Prime Factorization of Each Number

**Factor tree for 30:**

- Start by dividing 30 by 2 (the smallest prime number):
  \[
  30 \div 2 = 15
  \]
- Now, divide 15 by 3 (the next smallest prime number):
  \[
  15 \div 3 = 5
  \]
- 5 is a prime number, so we stop here.

So, the prime factorization of 30 is:
\[
30 = 2^1 \times 3^1 \times 5^1
\]

**Factor tree for 36:**

- Start by dividing 36 by 2:
  \[
  36 \div 2 = 18
  \]
- Now, divide 18 by 2:
  \[
  18 \div 2 = 9
  \]
- Divide 9 by 3:
  \[
  9 \div 3 = 3
  \]
- 3 is a prime number, so we stop here.

So, the prime factorization of 36 is:
\[
36 = 2^2 \times 3^2
\]

### Step 2: Identify the LCM

To find the LCM, take the highest power of each prime factor present in the factorization of both numbers:

- **For the prime factor 2:** The highest power is \(2^2\).
- **For the prime factor 3:** The highest power is \(3^2\).
- **For the prime factor 5:** The highest power is \(5^1\).

Now, multiply these together to find the LCM:
\[
\text{LCM} = 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = 180
\]

So, the LCM of 30 and 36 is **180**.

### Factor Trees Summary:

- **30:** 
\[
\begin{array}{ccc}
& 30 & \\
& / & \backslash \\
15 & & 2 \\
& / & \backslash \\
5 & & 3 \\
\end{array}
\]
- **36:**
\[
\begin{array}{ccc}
& 36 & \\
& / & \backslash \\
18 & & 2 \\
& / & \backslash \\
9 & & 2 \\
& / & \backslash \\
3 & & 3 \\
\end{array}
\]

Would you like more details, or do you have any questions?

---

### Further Practice Problems:

1. Find the LCM of 45 and 60 using the factor tree method.
2. Calculate the LCM of 14 and 28.
3. Determine the LCM of 48 and 72.
4. Find the prime factorization of 50 and 75, then compute the LCM.
5. Identify the LCM of 12, 18, and 24.
6. Calculate the LCM of 20 and 25.
7. Determine the LCM of 42 and 56.
8. Find the LCM of 9 and 12.

**Tip:** Always remember to take the highest power of each prime factor when finding the LCM!

Learn how to find the Least Common Multiple (LCM) of 30 and 36 using factor trees with a detailed step-by-step solution. Understand prime factorization, identify LCM using highest prime factors, and practice similar problems. Suitable for students in Grades 6-8.

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