AMC 10 · 2013 · #17
Grade 6 number-theoryPick an answer.
AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
"Exactly two" breaks into three separate pair problems: Alice-Beatrix, Alice-Claire, and Beatrix-Claire. Two friends share a day only when it is a common multiple of their cycles, so each pair meets on multiples of a least common multiple. Counting each pair, then removing the days when all three actually show up, gives the exactly-two total.
Turn each pair into an LCM
A shared day must fit both cycles, so each pair's gap is its LCM: 12, 15, and 20 days.
A shared visit day has to fit both friends' cycles at once, so it lands on their LCM.
A shared visit day has to fit both cycles at once, so it lands on their least common multiple.
▸ Why?
A day on both cycles is a multiple of each, so the first such day is their least common multiple.
▸ Why?
Each visit pattern repeats identically after its own period, so the meetings then repeat on that rhythm.
Count each pair's meetings
Divide 365 by each gap and drop the remainder: the pairs meet 30, 24, and 18 times.
The number of multiples of k up to 365 is just how many whole times k fits into 365.
6.NS.B.2Identify SubproblemsFind the all-three days
All three coincide only on multiples of LCM(3,4,5)=60, and 365 days hold 6 such days.
A day that suits all three cycles must be a multiple of their combined LCM, 60.
6.NS.B.4Identify SubproblemsRemove the triple days from each pair
Those 6 triple days sit in every pair total, so strip them from each: (30-6)+(24-6)+(18-6)=54.
A day only counts as exactly two if the third friend stays away, so the all-three days must be stripped out of every pair.
4.OA.A.3Change Focus Count The ComplementTwo friends meet on the LCM of their cycles, so count each pair's meetings and take out the days when all three show up.
- Turn each pair into an LCM
- Count each pair's meetings
- Find the all-three days
- Remove the triple days from each pair