Nina is putting bottles and cans in bags to take to a recycling facility. She puts bottles in bags of 12 and cans in bags of 15, with no partially filled bags. While counting up the cans and bottles, an employee at the recycling facility points out that Nina has the same number of each. What is the smallest number of bottles that she can recycle?
LCM(12,15) = 60
To find the smallest number of bottles that Nina can recycle, we need to find a common multiple of 12 and 15, as this will represent the smallest number of items she can have in both categories.
To find the common multiple, we can list out the multiples of both 12 and 15 until we find a number that appears in both lists.
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, ...
Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, 135, ...
From the lists, we can see that 60 is the smallest common multiple of 12 and 15. Thus, Nina can recycle the smallest number of bottles when she has 60 bottles.