Two bus routes - Math Central

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Question from kiyah, a student:

from 4:30 pm to 6:30 pm the route 1 bus stops every 12 min at the gym's bus stop. the route 2 bus stops there every 15 min. if both buses are now at the stop and schedule is kept, how long will it be before both buses will be at the stop again?

Kiyah,

The route 1 bus will stop at the gym in 12 minutes, 24 minutes, 36 minutes and so on. Every stop time is a multiple of 12. The route 2 bus will stop in 15 minutes, 30 minutes etc. Every stop time is a multiple of 15. You can just continue each list of times and look for the first time the same number appears in both lists. So you are looking for a common multiple of 12 and 15, in fact the smallest or least number that is a multiple of 12 and 15. We call the number you are looking for the least common multiple or LCM of 12 and 15.

The listing method can become tedious. There is another technique that depends on the prime factorization of the numbers.

12 = 2 × 2 × 3 and
15 = 3 × 5.

From this we can see that

if a number is a multiple of 12 its prime factorization contains at least two 2s and at least one 3
if a number is a multiple of 15 its prime factorization contains at least one 3 and at least one 5.

Hence if number is a multiple of 12 and 15 its prime factorization must contain at least two 2s, at least one 3 and at least one 5. What is the smallest such number?

Penny

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