Competition · AMC preparation · step 4 of 4
AMC 8 · 2017 · #24
Grade 6 number-theorycountingPick an answer.
AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The question asks for days with NO call — a textbook 'at least one' / 'none' situation, which is Tool #16 (Complement): count call days first, then subtract from 365. Tool #12 (Venn) is the natural picture for three overlapping sets A (multiples of 3), B (multiples of 4), C (multiples of 5); the Venn diagram makes inclusion-exclusion concrete — we add the three circles, subtract each pairwise overlap (multiples of lcm), then add back the triple overlap. Tool #5 (Pattern) covers the simple counting rule: the number of multiples of k in {1, …, 365} is just ⌊ 365/k ⌋.
Count each grandchild's call days
Each grandchild's call-days are the multiples of its interval up to 365, counted by ⌊365/k⌋: |A| = 121, |B| = 91, |C| = 73.
Counting multiples of a number in a range is the same Grade 4 'factors and multiples' skill, just applied three times.
4.OA.B.4Look For A PatternCount days two both call
Two grandchildren coincide only on multiples of their LCM (12, 15, 20): |A ∩ B| = 30, |A ∩ C| = 24, |B ∩ C| = 18.
Two events happen on the same day exactly when the day is in both circles of the Venn diagram — which is a multiple of their LCM (Grade 6 skill).
6.NS.B.4Draw A Venn DiagramCount days all three call
All three coincide only on multiples of lcm(3,4,5) = 60, the center of the Venn diagram: |A ∩ B ∩ C| = 6.
The triple-overlap days are multiples of the LCM of all three intervals — the deepest center of the Venn diagram.
6.NS.B.4Draw A Venn DiagramCombine by inclusion-exclusion
Inclusion-exclusion: add the circles, subtract the pairwise overlaps, add back the triple → 219 days with at least one call.
The +/-/+ pattern is just the Venn diagram bookkeeping: each region of the picture gets counted exactly once when you finish.
The number of days with at least one call equals the sum of the three single-set counts, minus the three pairwise-overlap counts, plus the triple-overlap count.
▸ Why?
Every day with at least one call belongs to exactly one of three non-overlapping groups — called by one, by two, or by all three grandchildren — so the correct total should count each such day exactly once.
▸ Why?
Simply adding the three single-set counts tallies a two-caller day twice and a three-caller day three times, so that raw sum overcounts every day that is shared.
▸ Why?
Subtracting the three pairwise-overlap counts takes a two-caller day away one extra time, dropping it from counted-twice back to counted-once.
▸ Why?
Those same subtractions strip a three-caller day from counted-three-times down to zero, so adding the triple-overlap count back once restores it to counted-once.
Subtract from 365
No-call days are the complement of call days: 365 - 219 = 146, choice (D).
'No call' is the complement of 'at least one call' — subtract from the total. Matches choice (D).
4.NBT.B.4Change Focus Count The ComplementThis AMC 8 problem only needs Grade 6 LCM (least common multiple) plus a Venn diagram you already know!
- Count each grandchild's call days
- Count days two both call
- Count days all three call
- Combine by inclusion-exclusion
- Subtract from 365
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