How do I find the greatest common factor of 54, 81, and 135? I don't understand the tree diagram either. :(
I usually start with the factors of the largest number. The obvious factor of 135 is 5. Bingo! 5 * 27 = 135
27 is also a factor of 81 and 54.
This site explains the factor tree.
http://www.mathsisfun.com/definitions/factor-tree.html
To find the greatest common factor (GCF) of three numbers, such as 54, 81, and 135, it's helpful to use a method called factoring. Here's how you can approach it step by step:
Step 1: Prime factorize each number. Prime factorization means expressing a number as a product of its prime factors.
Let's start with 54. Divide it by the smallest prime number, which is 2: 54 ÷ 2 = 27. Since 27 is not divisible by 2, move on to the next prime number, which is 3: 27 ÷ 3 = 9. Repeat the process once more: 9 ÷ 3 = 3. Now, we have fully prime factorized 54 as 2 × 3 × 3 × 3, or 2 × 3^3.
Next, let's prime factorize 81. Start dividing it by 3: 81 ÷ 3 = 27. Continue dividing by 3: 27 ÷ 3 = 9. Lastly, divide by 3 again: 9 ÷ 3 = 3. So, 81 can be written as 3 × 3 × 3 × 3, or 3^4.
Finally, let's prime factorize 135. Begin with dividing by 3: 135 ÷ 3 = 45. Divide 45 by 3: 45 ÷ 3 = 15. Divide 15 by 3: 15 ÷ 3 = 5. Hence, 135 can be written as 3 × 3 × 3 × 5, or 3^3 × 5.
Step 2: Make a tree diagram to find the shared prime factors. This diagram will show all the prime factors of each number, and the shared prime factors will be aligned vertically.
We'll start with the prime factors of 54: 2, 3, and 3. Write these down.
Next, we'll consider the prime factors of 81: 3, 3, 3, and 3. Write down these shared factors below the corresponding factors of 54.
Lastly, we'll list the prime factors of 135: 3, 3, 3, and 5. Again, write the shared factors below the corresponding factors of 54 and 81.
So far, the shared prime factors are: 3, 3, and 3.
Step 3: Multiply the shared prime factors. To find the GCF, we multiply the shared prime factors together: 3 × 3 × 3 = 27.
Therefore, the GCF of 54, 81, and 135 is 27.
I hope this step-by-step explanation helps you understand the process.