Greta is making fruit baskets. She has 40 apples and 30 oranges. Greta wants to make all the fruit baskets identical without having any pieces of fruit left over. What is the greatest number of fruit baskets Greta can make?

Answer 1

To determine the greatest number of identical fruit baskets that Greta can make using 40 apples and 30 oranges without any leftover fruit, we need to find the greatest common divisor (GCD) of the two quantities: 40 and 30.

First, we can find the GCD of 40 and 30 using the prime factorization method.

1. Find the prime factorization:

   • The prime factorization of 40 is \(2^3 \times 5^1\).
   • The prime factorization of 30 is \(2^1 \times 3^1 \times 5^1\).

2. Identify the common factors:

   • The common prime factors are \(2\) and \(5\).
   • The minimum exponent for \(2\) between the two factorizations is \(1\) (from 30).
   • The minimum exponent for \(5\) is \(1\) (from both).

3. Calculate the GCD: \[ \text{GCD} = 2^1 \times 5^1 = 2 \times 5 = 10. \]

Thus, the GCD of 40 and 30 is 10.

This means that the greatest number of identical fruit baskets Greta can make is \( \boxed{10} \).

Next, we can verify this by seeing how many pieces of fruit will be in each basket:

• If Greta makes 10 baskets, she can distribute:
  • Apples: \( \frac{40}{10} = 4 \) apples per basket,
  • Oranges: \( \frac{30}{10} = 3 \) oranges per basket.

Each basket will contain 4 apples and 3 oranges, with no fruits left over, confirming that 10 is indeed the greatest number of identical fruit baskets she can make.