Problem
Prime factorization two ways
72 is broken down to primes in two different ways. A factor tree on the left, a stack of short divisions on the right. Both finish at the same prime factorisation. Do the same for 54 in both ways.
Reasoning · Grade 5-1 Primes and Divisors
Your answer
Watch out The first split seems to matter, but the prime factorisation is unique.
How to solve
Strategy Look for a Pattern — The 72 example is a fully worked easier version of exactly the task I have, so I read the rule off it first: keep splitting until nothing splits any more. The factor tree is a drawing, so tool 1 does the work there; the short-division stack is the same information organised as a column instead of a tree, which is tool 15 — and comparing the two organisations is what makes the punchline visible. The punchline is a pattern: whichever split you start with, and whichever method you use, the collection of primes at the end is the same one.
1STEP 1
Read the stopping rule off the 72 example
The tree splits until every end is prime.
72 = 8 × 9 = (4 × 2) × (3 × 3) = (2 × 2 × 2) × (3 × 3)
Telling prime from composite is a Grade 4 skill: 8 has a factor pair besides 1 × 8, so it can still split; 3 has none, so it is finished.
4.OA.B.4Solve An Easier Related Problem2STEP 2
Method 1 — build a factor tree for 54
Splitting 54 as 6 and 9 ends at 2, 3, 3, 3.
54 = 6 × 9 = (2 × 3) × (3 × 3) = 2 × 3 × 3 × 3
Drawing it as a tree keeps track of the bookkeeping for you: whatever is written above always equals the product of the two things written below it, so the bottom row still multiplies back to 54.
4.OA.B.4Draw A Diagram3STEP 3
Start the tree differently and watch the ends stay the same
Splitting as 2 and 27 gives the same ends.
54 = 2 × 27 = 2 × 3 × 9 = 2 × 3 × 3 × 3
This is the pattern the page is really about: the shape of the tree is your choice, but the primes at the ends are not — the number itself decides them.
4.OA.B.4Look For A PatternStarting the factor tree a different way still ends at the very same collection of primes.
Why?
A number breaks into primes in only one way, so the route taken cannot change what the leaves turn out to be.
🧱Every number has one prime factorizationKeep splitting a number into factors and you always land on the same bag of primes, no matter which split you tried first.
Why?
Which two factors you split off first makes no difference to the product, so both trees describe the same multiplication.
🧱Associativity of multiplicationMultiplying three numbers, grouping any two first gives the same product.
4STEP 4
Method 2 — divide 54 by small primes, over and over
The division stack divides by 2, 3, 3 in turn.
54 ÷ 2 = 27, 27 ÷ 3 = 9, 9 ÷ 3 = 3
Each division is a small one a Grade 4 solver can do in their head, and the number shrinks every time, so the process is guaranteed to stop — it cannot shrink below 1.
4.NBT.B.6Organize Information In More Ways5STEP 5
Read the answer off the stack and compare the two methods
The divisors and last quotient give 2 × 3 × 3 × 3.
54 = 2 × 3 × 3 × 3
Two very different-looking routes ending at the same list is strong evidence that the list belongs to the number, not to the route.
4.OA.B.4Look For A Pattern6STEP 6
Write it in canonical form and check by multiplying back
Multiplying back gives 54.
54 = 2 × 3 × 3 × 3 = 2 × 3³, 2 × 27 = 54
Sorting and collecting is only a tidying-up move, but it is the move that makes 'the same factorization' something you can check at a glance instead of by comparing two drawings.
6.EE.A.1Organize Information In More WaysAnswer
2 × 3 × 3 × 3
2 × 27 = 54
Multiplying the answer back gives 2 × 3 × 3 × 3 = 54, so the size is right. Every factor listed really is prime — 2 and 3 have no divisors other than 1 and themselves — so the factorization is complete and none of the ends could split further. The digit-test check also agrees: 54 is even, so a 2 must appear, and its digit sum is 5 + 4 = 9, a multiple of 9, so at least two 3s are forced by that test. There is no digit-sum test for 27, so the exact count is settled by dividing instead: the third 3 and the 2 come from 54 ÷ 9 = 6 = 2 × 3. Comparing with the worked example is a further sanity check: 72 = 2³ × 3² and 54 = 2 × 3³ are both built only from 2s and 3s, which fits since 72 = 8 × 9 and 54 = 2 × 27 are both products of a power of 2 and a power of 3.
Takeaway
You can split a number any way you like, but you always end up with the same bag of primes — 54 is always one 2 and three 3s.
- Read the stopping rule off the 72 example
- Method 1 — build a factor tree for 54
- Start the tree differently and watch the ends stay the same
- Method 2 — divide 54 by small primes, over and over
- Read the answer off the stack and compare the two methods
- Write it in canonical form and check by multiplying back