To find the longest length of boards that can be cut from the given lengths of 24, 36, and 42 inches with no scrap wood left over, we need to calculate the greatest common divisor (GCD) of these lengths.
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First, we find the prime factorization of each length:
- 24: \( 2^3 \times 3^1 \)
- 36: \( 2^2 \times 3^2 \)
- 42: \( 2^1 \times 3^1 \times 7^1 \)
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Next, we identify the minimum power of each prime factor in all three factorizations:
- For the factor \(2\), the minimum power is \(1\) (from 42).
- For the factor \(3\), the minimum power is \(1\) (from both 24 and 42).
- The factor \(7\) does not appear in both 24 and 36, so we ignore it.
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Now, we multiply the lowest powers of all prime factors: \[ \text{GCD} = 2^1 \times 3^1 = 2 \times 3 = 6 \]
Thus, the longest length of boards that can be cut without leaving any scrap is 6 inches.
The answer is: 6 inches