One bus leaves every 12 minutes and another every 18 minutes. They leave together at 9:00. When do they next leave together?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
List the departure times for each bus.
2The strategy
You want the first time that is a multiple of both intervals. Listing the multiples of each and finding the first one they share is quicker than any formula at these sizes.
3The full solution
Multiples of 12: 12, 24, 36. Multiples of 18: 18, 36. The first shared one is 36 minutes, so 9:36.
Solution
The reliable way
List the minutes after 9:00 at which each bus leaves. The first bus goes at 12, 24, 36, 48; the second at 18, 36, 54. The first time both appear is 36 minutes, so they next leave together at 9:36. The transferable idea: two repeating cycles coincide at their least common multiple.
The elegant way
12 = 2 x 2 x 3 and 18 = 2 x 3 x 3, so the least common multiple is 2 x 2 x 3 x 3 = 36.
Why this is on the test: Coinciding cycles is the standard dressing for least common multiple, and reaching for the greatest common divisor instead is the reliable error.