To find the greatest length of fabric pieces that Mrs. Smith can cut from the bolts of material with lengths of 42, 63, and 105 inches, we need to determine the greatest common divisor (GCD) of these three lengths.
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Finding the prime factorization of each number:
- 42:
- \(42 = 2 \times 3 \times 7\)
- 63:
- \(63 = 3^2 \times 7\)
- 105:
- \(105 = 3 \times 5 \times 7\)
- 42:
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Identifying the common factors:
- The common factors among the three numbers are \(3\) and \(7\).
- The GCD is found by multiplying the lowest powers of all prime factors that appear in all factorizations.
- For \(3\), the lowest power is \(3^1\).
- For \(7\), the lowest power is \(7^1\).
- Hence, \(GCD = 3^1 \times 7^1 = 3 \times 7 = 21\).
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Conclusion: The greatest length of fabric pieces that Mrs. Smith can cut from the bolts of material with lengths of 42, 63, and 105 inches, such that none is left over, is 21 inches.
Thus, the answer is 21 inches.