### 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 need to find the least common multiple of $12$ and $16$ minutes, because that's how long it will take them to complete the same number of laps and return to the starting line.
To find the LCM, I can use the prime factorization method.
I write $12$ as $2^2 \cdot 3$ and $16$ as $2^4.$
Then I take the highest power of each prime factor that appears in either factorization.
So I have $2^4 \cdot 3$ as the LCM.
Multiplying, I get $48$ minutes as the LCM.
That means they will cross the starting line together every $48$ minutes after 12:15 PM.
To find the next time, I add $48$ minutes to 12:15 PM.
I can do this by adding $45$ minutes first, then $3$ more minutes.
Adding $45$ minutes to 12:15 PM gives me 1:00 PM.
Then adding $3$ more minutes gives me 1:03 PM.
So the next time they cross the starting line together is 1:03 PM.