Question

A supplier receives orders from 5 different buyers. Each buyer places their order only on a Monday. The first buyer places the order after every 2 weeks, the second buyer, after every 6 weeks, the third buyer, after every 8 weeks, the fourth buyer, every 4 weeks, and the fifth buyer, after every 3 weeks. It is known that on January 1st, which was a Monday, each of these five buyers placed an order with the supplier.
On how many occasions, in the same year, will these buyers place their orders together excluding the order placed on January 1st?

• A. 1
• B. 5
• C. 4
• D. 2
• E. 3

Answer 1

The Correct Option is D

To find out how many occasions these five buyers will place their orders together in the same year, excluding the order placed on January 1st, we need to determine when they will all place their orders on the same Monday again.

The key is to calculate the Least Common Multiple (LCM) of the intervals at which the buyers place their orders. The intervals are:

• Buyer 1: every 2 weeks
• Buyer 2: every 6 weeks
• Buyer 3: every 8 weeks
• Buyer 4: every 4 weeks
• Buyer 5: every 3 weeks

First, we find the LCM of these intervals:

1. \(2 = 2^1\)
2. \(3 = 3^1\)
3. \(4 = 2^2\)
4. \(6 = 2^1 \times 3^1\)
5. \(8 = 2^3\)

The LCM is found by taking the highest power of each prime number appearing in the factorizations:

• Highest power of \(2\) is \(2^3\)
• Highest power of \(3\) is \(3^1\)

Thus, the LCM is:

\(LCM = 2^3 \times 3^1 = 8 \times 3 = 24\)

This means all buyers will place their orders together every 24 weeks. Since the year starts on January 1st, itself a Monday, we count the number of 24-week intervals within the same year.

There are 52 weeks in a year. The number of complete 24-week cycles that fit into 52 weeks is calculated as:

\(\lfloor \frac{52}{24} \rfloor = 2\)

Therefore, excluding the order on January 1st, there will be 2 more occasions when all the buyers place their orders together. Thus, the correct answer is:

2

Answer 2

To solve this problem, we need to find the Least Common Multiple (LCM) of the buyers' order intervals. This will tell us after how many weeks all buyers will place orders together, excluding January 1st. 
The order intervals are:

• Buyer 1: 2 weeks
• Buyer 2: 6 weeks
• Buyer 3: 8 weeks
• Buyer 4: 4 weeks
• Buyer 5: 3 weeks

Now, we find the LCM of these numbers:

1. Prime factorization:
   • 2 = 2
   • 6 = 2 × 3
   • 8 = 2³
   • 4 = 2²
   • 3 = 3
2. Take the highest power of each prime:
   • For 2: 2³
   • For 3: 3¹
3. LCM = 2³ × 3 = 8 × 3 = 24 weeks

Thus, every 24 weeks, all buyers will place orders together. Next, calculate how many 24-week intervals fit in a year excluding January 1st:

1. There are 52 weeks in a year.
2. Ordering starts on January 1st, a Monday.
3. Calculate additional occasions: 52 ÷ 24 ≈ 2 (integer division).

This means, excluding January 1st, the orders occur together twice more in the same year. Therefore, the correct answer is 2.