To find the smallest number of textbooks that both Harrison's Publishing Incorporated and Douglas Associates could have printed while meeting their respective printing requirements, we need to determine the least common multiple (LCM) of the two printing quantities, which are 7 and 6.
-
Prime factorization:
- The prime factorization of 7 is \(7^1\).
- The prime factorization of 6 is \(2^1 \times 3^1\).
-
Finding the LCM: The LCM is found by taking the highest power of each prime that appears in the factorizations:
- For the prime \(2\), the highest power is \(2^1\).
- For the prime \(3\), the highest power is \(3^1\).
- For the prime \(7\), the highest power is \(7^1\).
Thus, the LCM is calculated as follows: \[ \text{LCM} = 2^1 \times 3^1 \times 7^1 = 2 \times 3 \times 7 \]
- Calculating the product:
- First, calculate \(2 \times 3\): \[ 2 \times 3 = 6 \]
- Then multiply this result by \(7\): \[ 6 \times 7 = 42 \]
Therefore, the least common multiple of 7 and 6 is 42.
- Conclusion: The smallest number of textbooks that each company could have printed is
\[ \boxed{42}. \]
This means Harrison's Publishing printed \( \frac{42}{7} = 6 \) sets of textbooks, and Douglas Associates printed \( \frac{42}{6} = 7 \) sets of textbooks, both resulting in a total of 42 textbooks.
1 answer