To determine the greatest number of vegetables that can be in each box, we need to find the greatest common divisor (GCD) of the total number of carrots and the total number of potatoes.
The farmer has:
- 90 carrots
- 120 potatoes
We start by finding the GCD of 90 and 120.
Step 1: Find the prime factorization of each number.
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The prime factorization of 90: \[ 90 = 2 \times 3^2 \times 5 \]
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The prime factorization of 120: \[ 120 = 2^3 \times 3 \times 5 \]
Step 2: Identify the common prime factors and their lowest powers.
- For \(2\): the lowest power is \(2^1\)
- For \(3\): the lowest power is \(3^1\)
- For \(5\): the lowest power is \(5^1\)
Step 3: Calculate the GCD.
The GCD is found by multiplying the lowest powers of the common prime factors: \[ \text{GCD} = 2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30 \]
Conclusion
Thus, the greatest number of vegetables that can be in each box is \( \boxed{30} \).