This worksheet helps students break down composite numbers like 30 and 72 into their prime factors using tree diagrams.

I will solve this prime factor tree worksheet by completing each tree to find the prime factors of the given numbers: 30, 81, 56, and 72.

I'll start with the first tree for the number 30.

First, look closely: The tree for 30 is partially filled. It shows that 30 is broken down into 5 and another number (represented by a blank box). Since 5 is a prime number, I need to find its pair factor. Then, I will break down that second number into its own prime factors.

Let me calculate the missing value in the first tree.
- 30 divided by 5 equals 6.
- So, the blank box next to 5 should be filled with 6.
- Now, I need to break down 6 into its prime factors, which are 2 and 3.

I will now move on to the second tree for the number 81.

Next, find information: I need to determine the prime factors of 81. I know that 81 is 9 times 9, and 9 is 3 times 3. So, 81 can be broken down as follows:
- 81 = 9 × 9
- 9 = 3 × 3
- Therefore, 81 = 3 × 3 × 3 × 3

The tree already shows 81 broken down into two boxes, one of which is 9. The other box must also be 9. Then, each 9 is broken down into 3 and 3.

Now, let's work on the third tree for the number 56.

Then, review the findings: For 56, I need to find its prime factors. I can start by dividing 56 by 2, which gives 28. Then, 28 divided by 2 is 14, and 14 divided by 2 is 7. Since 7 is a prime number, the prime factors of 56 are 2, 2, 2, and 7.

The tree shows 56 broken down into two boxes. One possible way is to break it down into 2 and 28. Then, 28 can be broken down into 2 and 14, and 14 into 2 and 7.

Finally, I will solve the fourth tree for the number 72.

First, look closely: The tree for 72 is broken down into two boxes, and one of those boxes is further broken down into two more boxes, one of which is then broken down into two circles. I need to find the prime factors of 72.

I know that 72 can be broken down as follows:
- 72 = 8 × 9
- 8 = 2 × 2 × 2
- 9 = 3 × 3
- Therefore, 72 = 2 × 2 × 2 × 3 × 3

Another way to break it down is:
- 72 = 2 × 36
- 36 = 2 × 18
- 18 = 2 × 9
- 9 = 3 × 3

So, the prime factors are 2, 2, 2, 3, and 3.

Based on the structure of the tree, I can fill in the blanks accordingly.

To summarize, here are the completed prime factor trees:

1. For 30:
- 30 = 5 × 6
- 6 = 2 × 3
- Prime factors: 2, 3, 5

2. For 81:
- 81 = 9 × 9
- 9 = 3 × 3
- Prime factors: 3, 3, 3, 3

3. For 56:
- 56 = 2 × 28
- 28 = 2 × 14
- 14 = 2 × 7
- Prime factors: 2, 2, 2, 7

4. For 72:
- 72 = 8 × 9
- 8 = 2 × 2 × 2
- 9 = 3 × 3
- Prime factors: 2, 2, 2, 3, 3

I have now completed all the prime factor trees as required.