# Which is the most correct factor tree?

I have been working through some factoring problems that include factor trees.

Problem - using a factor tree, express 54 as a product of prime factors.

My answer:

54

2*27

2*3*9

2*3*3*3

Textbook answer:

54

6*9

2*3*3*3

Although they are both correct, I am curious why the textbook excluded using the prime factor of 2 throughout the tree. Is there a rule or convention I am not aware of that made my answer different from the textbook?

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The textbook approach splits 54, which has 4 prime factors (counting multiplicity) into two pairs, so has one less step than yours. Your approach means you just have to find one prime factor per step. Both work fine, you get to decide which is easier for you, and don't have to be consistent. –  Ross Millikan Oct 23 '12 at 13:39
Thanks, so the only thing that matters is the prime factors that come from the end of the process, regardless of how the tree is factored. I think I will stay consistent with my slightly longer method. The reason being is that I start factoring using the lowest prime factor, which for me seems appropriate for consistency. –  InHouse Oct 23 '12 at 13:48
The Fundamental Theorem of Arithmetic (en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic) guarantees that you will get the same prime factors in the end no matter how you choose to decompose the number. –  Austin Mohr Oct 23 '12 at 13:58
The choice of $6\cdot9$ for the first factorization may simply reflect the fact that we all learned that product in grade school, so it should leap to the mind on seeing $54$; we didn’t learn $2\cdot27$, so for most people it would have to be computed by actually dividing $54$ by $2$. –  Brian M. Scott Oct 23 '12 at 14:56

## 2 Answers

You can do it in any way you want. Prime factorisation is unique (the fundamental theorem of arithmetic), so as long as you don't use any factorisations that aren't true in your working, you'll always end up with the same prime factorisation.

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Both of them is correct. There isn't any rules to factor a number. However, the answer should be correct.

$54 = (2)\times(3)^3$

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