Claire bought 180 candies and 140 pens for goody bags for her birthday. What is the largest number of goody bags that Claire can make so that each goody bag has the same number of candies and the same number of pens? (All candies and pens should be used.)
The Correct Answer and Explanation is:
To find the largest number of goody bags that Claire can make so that each bag has the same number of candies and the same number of pens, we need to determine the greatest common divisor (GCD) of 180 and 140.
Step-by-step:
- Find the GCD of 180 and 140
Start by finding the prime factorizations: - 180 = 2 × 2 × 3 × 3 × 5 = 2² × 3² × 5
- 140 = 2 × 2 × 5 × 7 = 2² × 5 × 7
Now identify the common prime factors:
- Both numbers share 2² and 5
Multiply the shared prime factors:
- GCD = 2² × 5 = 4 × 5 = 20
Final Answer:
20 goody bags
Explanation
Claire wants to divide her 180 candies and 140 pens into identical goody bags. This means each bag must contain the same number of candies and the same number of pens. Also, since she wants to use up all of the items without any leftovers, the number of goody bags must divide both 180 and 140 evenly.
To solve this, we look for the greatest common divisor (GCD) of 180 and 140. The GCD is the largest number that divides both numbers exactly. This tells us how many groups (goody bags) we can create where each group is the same and nothing is left over.
We begin by breaking down each number into its prime factors. For 180, the prime factorization is 2 × 2 × 3 × 3 × 5. For 140, it is 2 × 2 × 5 × 7. Next, we find the factors that both numbers share. Both 180 and 140 include 2 × 2 and 5 in their factorizations. These shared factors are important because they represent the greatest number of identical parts both quantities can be divided into.
When we multiply the shared prime factors—2 × 2 = 4 and then 4 × 5 = 20—we get the GCD. This means Claire can make a maximum of 20 goody bags. Each bag will have 180 ÷ 20 = 9 candies and 140 ÷ 20 = 7 pens.
By using the GCD, we ensure that every goody bag has the same contents and that all 180 candies and all 140 pens are used up perfectly.
