Every 6th customer at the flower shop receives a free rose, and every 9th customer receives a free lily. Which customer will be the first to receive a free rose and a free lily?
The correct answer and explanation is:
Correct Answer: 18th customer
Explanation (300 words):
This is a classic least common multiple (LCM) problem, often seen in math questions involving repeating patterns or schedules.
The problem states:
- Every 6th customer gets a free rose.
- Every 9th customer gets a free lily.
We are asked to find the first customer who receives both a rose and a lily. This customer must be both a multiple of 6 and a multiple of 9.
To solve this, we need to find the Least Common Multiple (LCM) of 6 and 9.
Step 1: Prime factorization
- 6 = 2 × 3
- 9 = 3 × 3
Step 2: Take the highest power of each prime
- 2¹ (from 6)
- 3² (from 9)
Now multiply these together:
- LCM = 2¹ × 3² = 2 × 9 = 18
So, the 18th customer is the first person to receive both a free rose and a free lily.
Why LCM Works Here:
The LCM of two numbers is the smallest number that both original numbers divide into evenly. In this case:
- Every 6th customer = 6, 12, 18, 24, …
- Every 9th customer = 9, 18, 27, 36, …
The first number common to both lists is 18.
Real-World Meaning:
If you work at or manage a flower shop, understanding patterns like this helps with planning promotions or inventory. For example, by the time you’ve served 18 customers, you should have at least one rose and one lily ready for the lucky customer who qualifies for both freebies.
Therefore, the 18th customer will be the first to receive both a free rose and a free lily.