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

Question

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

9
45
5
30

GPT-4o mini · Answer

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**.