Question
A restaurant receives a shipment of onions every 14 days, a shipment of salad greens every 6 days, and a shipment of potatoes every 8 days. The restaurant received shipments of all three items today. Use the dropdown menus to correctly complete both statements.
The next time that the restaurant will receive shipments of both onions and salad greens is in (Select) days.
The next time that the restaurant will receive shipments of both onions and potatoes is in (Select) days.
The next time that the restaurant will receive shipments of both onions and salad greens is in (Select) days.
The next time that the restaurant will receive shipments of both onions and potatoes is in (Select) days.
Answers
GPT-4o mini
To find out when the restaurant will receive shipments of both items, we need to determine the least common multiple (LCM) of the intervals for each pair of items.
1. **Onions and Salad Greens**:
- Onions are shipped every 14 days.
- Salad greens are shipped every 6 days.
To find the LCM of 14 and 6:
- The prime factorization of 14 is \(2 \times 7\).
- The prime factorization of 6 is \(2 \times 3\).
- The LCM is found by taking the highest power of each prime factor:
- \(2^1\), \(3^1\), \(7^1\).
- Thus, \(LCM = 2^1 \times 3^1 \times 7^1 = 2 \times 3 \times 7 = 42\).
Therefore, the next time the restaurant will receive shipments of both onions and salad greens is in **42 days**.
2. **Onions and Potatoes**:
- Onions are shipped every 14 days.
- Potatoes are shipped every 8 days.
To find the LCM of 14 and 8:
- The prime factorization of 14 is \(2 \times 7\).
- The prime factorization of 8 is \(2^3\).
- The LCM is found by taking the highest power of each prime factor:
- \(2^3\), \(7^1\).
- Thus, \(LCM = 2^3 \times 7^1 = 8 \times 7 = 56\).
Therefore, the next time the restaurant will receive shipments of both onions and potatoes is in **56 days**.
To complete the statements:
The next time that the restaurant will receive shipments of both onions and salad greens is in **42** days.
The next time that the restaurant will receive shipments of both onions and potatoes is in **56** days.
1. **Onions and Salad Greens**:
- Onions are shipped every 14 days.
- Salad greens are shipped every 6 days.
To find the LCM of 14 and 6:
- The prime factorization of 14 is \(2 \times 7\).
- The prime factorization of 6 is \(2 \times 3\).
- The LCM is found by taking the highest power of each prime factor:
- \(2^1\), \(3^1\), \(7^1\).
- Thus, \(LCM = 2^1 \times 3^1 \times 7^1 = 2 \times 3 \times 7 = 42\).
Therefore, the next time the restaurant will receive shipments of both onions and salad greens is in **42 days**.
2. **Onions and Potatoes**:
- Onions are shipped every 14 days.
- Potatoes are shipped every 8 days.
To find the LCM of 14 and 8:
- The prime factorization of 14 is \(2 \times 7\).
- The prime factorization of 8 is \(2^3\).
- The LCM is found by taking the highest power of each prime factor:
- \(2^3\), \(7^1\).
- Thus, \(LCM = 2^3 \times 7^1 = 8 \times 7 = 56\).
Therefore, the next time the restaurant will receive shipments of both onions and potatoes is in **56 days**.
To complete the statements:
The next time that the restaurant will receive shipments of both onions and salad greens is in **42** days.
The next time that the restaurant will receive shipments of both onions and potatoes is in **56** days.