How do you find a gcf using factor trees?

Answer 1

Example: 30 and 42

30

15,2

5,3,2

42

21,2

7,3,2

2 x 3 x 5 = 30

2 x 3 x 7 = 42

Select the common factors.

2 x 3 = 6, the GCF

Combine the factors, eliminating duplicates.

2 x 3 x 5 x 7 = 210, the LCM

Answer 2

The factor tree is used to find the prime factorisation of the numbers. The greatest common factor is then the product of every common prime across the numbers; if a prime occurs more than once in all the numbers, then it is included more than once.

For example the gcf(60, 72):

The trees are written with branches that multiply to the numbers from which they fork:

_________60______________________________72_________________

________/___\____________________________/___\________________

_______/_____\__________________________/_____\_______________

______4_____15_______________________8_______9______________

_____/_\____/__\______________________/_\______/_\_____________

____2__2__3___3___________________4___2____3__3____________

___________________________________/_\________________________

__________________________________2__2_______________________

(Note that 60 = 4 × 15, 8 = 4 × 2, etc. The actual factorisations down the tree do not matter as long as each number is split into a factor pair. I've had to use the underscore character (_) to keep the columns aligned as spaces get trimmed in answers - think of them like lines on a piece of paper.)

When the number is a prime, it is the end of a branch. The prime factorisation of the number is then read from the ends of all the branches:

60 = (2) × (2) × (3) × 5

72 = (2) × (2) × 2 × (3) × 3

The common primes I have enclosed in parentheses and are 2 × 2 × 3 = 12.

Thus gcf(60, 72) = 12.

If the prime factorisation is written in power format, the gcf is the product of the common primes to their lowest power. For the above, this becomes:

60 = 2² × 3 × 5

72 = 2³ × 3²

The common primes are 2 and 3. The lowest power of 2 is 2² and the lowest power of 3 is 3¹ = 3. Thus gcf = 2² × 3 = 12.

Answer 3

The factor tree is just a way of notating the process of finding the prime factorization.

Example: 30 and 42

30

15,2

5,3,2

42

21,2

7,3,2

2 x 3 x 5 = 30

2 x 3 x 7 = 42

Select the common factors.

2 x 3 = 6, the GCF

Answer 4

Example: 30 and 42

30

15,2

5,3,2

2 x 3 x 5 = 30

42

21,2

7,3,2

2 x 3 x 7 = 42

Select the common factors.

2 x 3 = 6, the GCF

Answer 5

Factor trees give you the prime factorization.

Comparing prime factorizations gives you the GCF.

Example: 30 and 42

30

15,2

5,3,2

42

21,2

7,3,2

2 x 3 x 5 = 30

2 x 3 x 7 = 42

Select the common factors.

2 x 3 = 6, the GCF

Answer 6

Example: 30 and 42

30

15,2

5,3,2

42

21,2

7,3,2

Select the common factors.

2 x 3 = 6, the GCF