To reduce the fraction \( \frac{180}{315} \), we need to find the greatest common factor (GCF) of the numerator (180) and the denominator (315).
First, let's find the prime factorization of both numbers:
For 180:
- Divide by 2: \( 180 \div 2 = 90 \)
- Divide by 2: \( 90 \div 2 = 45 \)
- Divide by 3: \( 45 \div 3 = 15 \)
- Divide by 3: \( 15 \div 3 = 5 \)
- 5 is prime.
So, the prime factorization of 180 is: \[ 180 = 2^2 \times 3^2 \times 5^1 \]
For 315:
- Divide by 3: \( 315 \div 3 = 105 \)
- Divide by 3: \( 105 \div 3 = 35 \)
- Divide by 5: \( 35 \div 5 = 7 \)
- 7 is prime.
So, the prime factorization of 315 is: \[ 315 = 3^2 \times 5^1 \times 7^1 \]
Now, we can find the GCF by taking the lowest powers of all the prime factors that appear in both factorizations:
- For \( 3 \): Minimum exponent is \( 2 \) (from both 180 and 315).
- For \( 5 \): Minimum exponent is \( 1 \) (from both 180 and 315).
- \( 2 \) and \( 7 \) do not appear in both factorizations.
Thus, the GCF of 180 and 315 is: \[ GCF = 3^2 \times 5^1 = 9 \times 5 = 45 \]
Therefore, Katy should use 45 to reduce the fraction \( \frac{180}{315} \) in one step.
The greatest common factor that Katy should use to reduce the fraction is 45.