FREE Printable Factor Tree Worksheets [PDFs] Brighterly.com - Free Printable

Let’s solve each prime factor tree step by step.

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First Tree: 30

We start with 30. One branch is already given: 5.

So, 30 ÷ 5 = 6 → the other top box is 6.

Now break down 6:
6 can be split into 2 and 3 (both primes).
So the two bottom circles under 6 are 2 and 3.

Final factors for 30: 5, 2, 3 — all primes ✔

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Second Tree: 81

One branch is given: 9.

So, 81 ÷ 9 = 9 → the other top box is 9.

Now break down each 9:

Each 9 splits into 3 × 3 (since 3 is prime).

So under the left 9: two circles → 3 and 3
Under the right 9: two circles → 3 and 3

Final factors for 81: 3, 3, 3, 3 — all primes ✔

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Third Tree: 56

No branches given yet. Let’s pick a pair of factors.

56 can be split as 7 × 8 (7 is prime, so we’ll leave it as one circle).

So left circle: 7, right box: 8

Now break down 8:
8 = 2 × 4 → but 4 isn’t prime, so keep breaking.

Better to do: 8 = 2 × 2 × 2

But in tree form:
From 8, split into 2 and 4 → then 4 splits into 2 and 2.

Wait — looking at the structure:
The tree shows:
- Top: 56
- Left: empty circle (prime)
- Right: box → which splits into two circles → one of those circles splits again into two more circles.

That means:
56 → [prime] and [composite]
Composite breaks into two numbers → one of them breaks again.

Try: 56 = 7 × 8 → 7 is prime (left circle), 8 goes to right box.

Then 8 → 2 and 4 → 2 is prime (first circle under 8), 4 goes to next box.

Then 4 → 2 and 2 (last two circles).

Perfect!

So:
- Left circle from 56: 7
- Right box from 56: 8
- Under 8: left circle = 2, right box = 4
- Under 4: two circles = 2 and 2

All final leaves: 7, 2, 2, 2 — all primes ✔

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Fourth Tree: 72

Structure:
Top: 72
Splits into two boxes (both composite likely)
Left box splits into two circles (primes)
Right box splits into one circle and another box → that box splits into two circles.

Let’s find factors.

Try: 72 = 8 × 9

Break 8: 2 × 4 → 4 → 2×2 → too many levels? Wait, let’s map to structure.

Actually, better:
72 = 6 × 12? Or 8 × 9?

Try 8 and 9:

Left box: 8 → splits into two primes? 8 = 2 × 4 → no, 4 not prime. So maybe not.

Wait — look at structure:

Left side: box → two circles → both must be prime → so left box must be product of two primes.

Right side: box → one circle + one box → that last box → two circles → so right box = prime × (prime × prime)

So total: 72 = (p1 × p2) × (p3 × p4 × p5)? No — wait, count leaf nodes.

Left branch: 2 leaves
Right branch: 3 leaves (one direct circle, plus two from the sub-box)
Total: 5 prime factors? But 72 = 2³ × 3² → five prime factors: 2,2,2,3,3 → yes!

So assign:

Left box: say 8 = 2 × 2 × 2? But structure only allows two children for left box → so left box must be 4 or 6 or 8? Wait.

If left box has two prime children → possible values: 4 (2×2), 6 (2×3), 9 (3×3), etc.

Try left box = 8? But 8 doesn't split into two primes directly unless we allow intermediate steps — but in this tree, left box splits directly into two circles → so must be semiprime (product of two primes).

Similarly, right box splits into one circle (prime) and one box (which splits into two primes) → so right box = prime × (prime × prime) = three primes multiplied.

So overall: 72 = (two primes) × (three primes) → matches 2×2×2×3×3.

Assign:

Left box: 4 = 2 × 2 → circles: 2, 2

Right box: 18 = 2 × 9 → but 9 = 3×3 → perfect.

So right box = 18 → splits into circle: 2, and box: 9

Then 9 splits into 3, 3

Check: 4 × 18 = 72 → yes.

Prime factors: 2,2,2,3,3 → correct.

Alternatively, could swap: left box = 6 = 2×3, right box = 12 = 3×4 → but 4 needs to break → same idea.

But let’s stick with above.

So:

- Left box from 72: 4 → circles: 2, 2
- Right box from 72: 18 → circle: 2, box: 9
- Box 9 → circles: 3, 3

Wait — but 4 × 18 = 72, and 18 = 2 × 9, 9=3×3 → factors: 2,2,2,3,3 → good.

But actually, 72 = 8 × 9 might fit better visually?

Try:

Left box: 8 → but 8 can’t split into two primes directly → unless we force 2 and 4, but 4 isn’t prime → invalid for direct split.

So our first assignment is correct.

Another option: left box = 6 = 2×3, right box = 12 = 3×4 → then 4=2×2.

So:

Left box: 6 → circles: 2, 3

Right box: 12 → circle: 3, box: 4

Box 4 → circles: 2, 2

Factors: 2,3,3,2,2 → same set → also valid.

Which one to choose? Either works. Let’s pick the second because 6 and 12 are common factors.

But let’s verify structure:

In the image, right side of 72: box → one circle and one box → then that box → two circles.

So if right box is 12, and it splits into 3 (circle) and 4 (box), then 4 splits into 2 and 2 → fits perfectly.

Left box: 6 → splits into 2 and 3 → both circles → fits.

Yes.

So final assignments:

For 72:
- Left box: 6 → circles: 2, 3
- Right box: 12 → circle: 3, box: 4
- Box 4 → circles: 2, 2

All primes: 2,3,3,2,2 → sorted: 2,2,2,3,3 → correct.

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Now compile all answers:

Tree 1 (30):
- Missing box: 6
- Bottom circles under 6: 2, 3

Tree 2 (81):
- Missing top box: 9
- Each 9 splits into: 3, 3

Tree 3 (56):
- Left circle: 7
- Right box: 8
- Under 8: left circle: 2, right box: 4
- Under 4: 2, 2

Tree 4 (72):
- Left box: 6 → circles: 2, 3
- Right box: 12 → circle: 3, box: 4
- Box 4: 2, 2

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Final Answer:
For 30: missing box is 6; its children are 2 and 3.
For 81: missing top box is 9; each 9 splits into 3 and 3.
For 56: left circle is 7; right box is 8; under 8: 2 and 4; under 4: 2 and 2.
For 72: left box is 6 (children 2, 3); right box is 12 (child 3 and box 4); box 4 has children 2 and 2.