The manager of a store wants to have a sales promotion where she gives prizes to the first several people through the door. She wants each prize to be an identical gift bag with some pins, ornaments, and mugs in it. The manager has 240 pins, 360 ornaments, and 540 mugs to put into the gift bags and wants to use all the items they have.

Question

If the manager wants to give prizes to as many people as possible, how many people will get gifts?

240
15
1,140
60

Answer 1

To determine how many identical gift bags can be made with the items available, we need to find the greatest common divisor (GCD) of the quantities of each item: 240 pins, 360 ornaments, and 540 mugs.

First, let's find the GCD of these three numbers.

1. Find the prime factorization of each number:

   • $240=2^4\times 3^1\times 5^1$
   • $360=2^3\times 3^2\times 5^1$
   • $540=2^2\times 3^3\times 5^1$

2. Identify the lowest power of each prime factor:

   • For $2$: the minimum power is $2^2$.
   • For $3$: the minimum power is $3^1$.
   • For $5$: the minimum power is $5^1$.

3. Calculate the GCD: $$\text{GCD}=2^2\times 3^1\times 5^1=4\times 3\times 5=60$$

Thus, the manager can make 60 identical gift bags.

4. Calculating the number of people that will receive gifts: Since the gifts are identical and formed into 60 bags, 60 people will receive gifts.

Hence, the number of people who will get gifts is 60.