To determine the greatest number of events that can be held with the specified number of students from both Allenville High and Kinley High, we need to find the greatest common divisor (GCD) of the numbers of students from both schools, which are 20 from Allenville High and 12 from Kinley High.
Let's start by finding the GCD of 20 and 12 using the prime factorization method.
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Prime factorization of 20:
- 20 can be factored into \( 2^2 \times 5^1 \).
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Prime factorization of 12:
- 12 can be factored into \( 2^2 \times 3^1 \).
Next, we identify the common prime factors and their lowest powers:
- The only common prime factor is \( 2 \).
- The lowest power of \( 2 \) in both factorizations is \( 2^2 \).
Thus, the GCD can be computed as: \[ \text{GCD}(20, 12) = 2^2 = 4. \]
Now, let's verify this by using the Euclidean algorithm:
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Divide 20 by 12: \[ 20 \div 12 = 1 \quad \text{(remainder } 8\text{)} \] Thus, we have: \[ 20 = 12 \times 1 + 8. \]
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Next, divide 12 by the remainder (8): \[ 12 \div 8 = 1 \quad \text{(remainder } 4\text{)} \] Therefore: \[ 12 = 8 \times 1 + 4. \]
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Then, divide 8 by the remainder (4): \[ 8 \div 4 = 2 \quad \text{(remainder } 0\text{)} \] Thus: \[ 8 = 4 \times 2 + 0. \]
When the remainder reaches 0, the GCD is the last non-zero remainder, which is 4.
Conclusion: The greatest number of events the commission can hold is \( \boxed{4} \). This means that in each event, 5 students from Allenville will pair with 3 students from Kinley, ensuring that all students are utilized without any overlaps.
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