AMC 10 · 2005 · #22
Grade 6 arithmeticPick an answer.
AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
A number shared by both lists must be a multiple of 4 and of 6 at the same time, so it must be a multiple of their least common multiple, 12. Tool #4 names any such shared number 12k, turning "how many common elements" into "how many whole numbers k work." Tool #7 splits the job into two smaller questions — find the biggest number each list reaches — and Tool #14 picks the tighter of the two ceilings, which is the only one that actually limits the count. Then counting the valid k is one division.
How far each list reaches
Each list ends at its 2005th multiple: S reaches 8020 and T reaches 12030.
The 2005th multiple of a number is just that number times 2005.
4.OA.B.4Identify SubproblemsA shared number is a multiple of 12
A number in both lists is a multiple of 4 and of 6, hence a multiple of 12 — call it 12k.
To sit in both lists a number must be built from 4s and from 6s at once, and 12 is the smallest number that is both.
To sit in both lists a number must be built from both step sizes at once, so it is a multiple of their least common multiple.
▸ Why?
A common multiple must contain both numbers' primes, and the smallest such number is that multiple.
▸ Why?
Every number has one prime recipe, so the two requirements combine with nothing counted twice.
Which ceiling actually limits it
Both ceilings must hold, and 8020 is the smaller one, so the binding condition is 12k ≤ 8020.
S stops sooner than T, so a shared number can only go as high as S allows.
6.EE.B.8Extreme PrincipleCount the valid multiples
Since k is a whole number and 8020 ÷ 12 = 668.33…, k runs from 1 to 668.
Each whole number k from 1 up to the cutoff gives exactly one shared number 12k.
6.NS.B.2Introduce A VariableNumbers in both lists must be multiples of the LCM (12), so count the multiples of 12 that fit under the shorter list's ceiling.
- How far each list reaches
- A shared number is a multiple of 12
- Which ceiling actually limits it
- Count the valid multiples