To find the greatest common factor (GCF) of the numbers 27 and 270, we can start by determining the prime factorization of each number.
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Prime factorization of 27:
- 27 can be factored as \(3 \times 3 \times 3\) or \(3^3\).
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Prime factorization of 270:
- 270 can be factored step by step:
- First, divide by 2: \(270 ÷ 2 = 135\).
- Then, divide 135 by 3: \(135 ÷ 3 = 45\).
- Then, divide 45 by 3: \(45 ÷ 3 = 15\).
- Finally, divide 15 by 3: \(15 ÷ 3 = 5\).
- So, \(270 = 2 \times 3^3 \times 5\).
- 270 can be factored step by step:
Now, the prime factorizations are:
- \(27 = 3^3\)
- \(270 = 2 \times 3^3 \times 5\)
To find the GCF, we take the lowest power of all the prime factors that appear in both factorizations.
The only common prime factor is \(3\), and the lowest power of \(3\) in both factorizations is \(3^3\).
Thus, the GCF of 27 and 270 is: \[ \text{GCF} = 3^3 = 27 \] So, the greatest common factor of 27 and 270 is 27.