AMC 8 · 2017 · #24

Grade 6 number-theorycounting

Archetype Modular Cycle Frequency Count └ Sub-archetype Floor Count Across Residue Bins

lcmmultiplesmodular-arithmeticcomplementary-counting complementary-countingpattern-recognition ↑ Prerequisites: lcmmultiples

📏 Long solution 💡 4 insights

Problem

Mrs. Sanders has three grandchildren who call her on fixed schedules: one every 3 days, one every 4 days, one every 5 days. All three called on December 31, 2016 (day 0), so within the next year — the 365 days of 2017 — the grandchildren call on days that are multiples of 3, 4, and 5 respectively. Count the days during 2017 on which she received NO phone call from any grandchild.

Pick an answer.

(A)

(B)

(C)

144

(D)

146

(E)

152

AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Change Focus / Count the Complement

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 ⌋.

1STEP 1

Each grandchild's call-days are the multiples of its interval up to 365, counted by ⌊365/k⌋: |A| = 121, |B| = 91, |C| = 73.

|A| = ⌊

\frac{365}{3}

⌋ = 121, |B| = ⌊

\frac{365}{4}

⌋ = 91, |C| = ⌊

\frac{365}{5}

⌋ = 73

2STEP 2

Two grandchildren coincide only on multiples of their LCM (12, 15, 20): |A ∩ B| = 30, |A ∩ C| = 24, |B ∩ C| = 18.

|A ∩ B| = ⌊

\frac{365}{12}

⌋ = 30, |A ∩ C| = ⌊

\frac{365}{15}

⌋ = 24, |B ∩ C| = ⌊

\frac{365}{20}

⌋ = 18

3STEP 3

All three coincide only on multiples of lcm(3,4,5) = 60, the center of the Venn diagram: |A ∩ B ∩ C| = 6.

|A ∩ B ∩ C| = ⌊

\frac{365}{60}

⌋ = 6

4STEP 4

Inclusion-exclusion: add the circles, subtract the pairwise overlaps, add back the triple → 219 days with at least one call.

|A ∪ B ∪ C| = (121 + 91 + 73) - (30 + 24 + 18) + 6 = 285 - 72 + 6 = 219

5STEP 5

No-call days are the complement of call days: 365 - 219 = 146, choice (D).

365 - 219 = 146 → (D)

Answer

146

Sanity check via the LCM(3,4,5) = 60-day cycle. In every 60-day block, the count of call days is 20 + 15 + 12 - 5 - 4 - 3 + 1 = 36, so 24 days per cycle have no call — a 40% no-call rate. Estimating: 365 ×

\frac{24}{60}

= 146 exactly. The result matches and lands on choice (D), with 146 comfortably inside the 78–152 option spread.

💡Key takeaway

This AMC 8 problem only needs Grade 6 LCM (least common multiple) plus a Venn diagram you already know!

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