Question

Katy wants to reduce 180/315, but she wants to do it in one step.

Which of the following is the greatest common factor that she should use to reduce this fraction?

9
45
5
30

Answer 1

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:**
1. Divide by 2: \( 180 \div 2 = 90 \)
2. Divide by 2: \( 90 \div 2 = 45 \)
3. Divide by 3: \( 45 \div 3 = 15 \)
4. Divide by 3: \( 15 \div 3 = 5 \)
5. 5 is prime.

So, the prime factorization of 180 is:
\[
180 = 2^2 \times 3^2 \times 5^1
\]

**For 315:**
1. Divide by 3: \( 315 \div 3 = 105 \)
2. Divide by 3: \( 105 \div 3 = 35 \)
3. Divide by 5: \( 35 \div 5 = 7 \)
4. 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**.