### Instruction: Two bicycle racers cross the starting line on a circular track at 12:15 PM. One cyclist takes $12$ minutes per lap, while the other completes a lap every $16$ minutes. Assuming their speeds remain constant, what is the next time they cross the starting line together?  Your answer should be in the form $h:m_1m_2,$ like 3:45.### Response: I want to find the time interval between their crossings at the starting line, and then add that to 12:15 PM to get the next time they cross.
To find the time interval, I need to find the least common multiple of 12 and 16, which is the smallest number of minutes that both cyclists can complete an integer number of laps in.
I can use the prime factorization method to find the LCM.
12 = 2 x 2 x 3 and 16 = 2 x 2 x 2 x 2.
The LCM is the product of the highest power of each prime factor, so LCM(12, 16) = 2 x 2 x 2 x 2 x 3 = 48.
This means that every 48 minutes, both cyclists cross the starting line together.
So the next time they cross is 48 minutes after 12:15 PM, which is 1:03 PM.