AMC 8 · 2017 · #24

Grade 6 number-theorycounting

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

📏 Long solution 💡 4 insights

Problem

Mrs. Sanders has three grandchildren, who call her regularly. One calls her every three days, one calls her every four days, and one calls her every five days. All three called her on December 31, 2016. On how many days during the next year did she not receive a phone call from any of her grandchildren?

$\textbf{(A) }78\qquad\textbf{(B) }80\qquad\textbf{(C) }144\qquad\textbf{(D) }146\qquad\textbf{(E) }152$

Pick an answer.

(A)

(B)

(C)

144

(D)

146

(E)

152

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Toolkit + CCSS Solution

Understand

Restated: 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.

Givens: Year $2017$ has $365$ days ($2017$ is not a leap year); Grandchild $A$ calls on every multiple of $3$ (days $3, 6, 9, \ldots$); Grandchild $B$ calls on every multiple of $4$ (days $4, 8, 12, \ldots$); Grandchild $C$ calls on every multiple of $5$ (days $5, 10, 15, \ldots$); Answer choices: (A) $78$, (B) $80$, (C) $144$, (D) $146$, (E) $152$

Unknowns: The number of days in $\{1, 2, \ldots, 365\}$ that are NOT a multiple of $3$, $4$, or $5$

Understand

Plan

Primary tool: #16 Change Focus / Count the Complement

Secondary: #12 Draw a Venn Diagram, #5 Look for a Pattern

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 $\text{lcm}$), then add back the triple overlap. Tool #5 (Pattern) covers the simple counting rule: the number of multiples of $k$ in $\{1, \ldots, 365\}$ is just $\lfloor 365/k \rfloor$.

Execute — Answer: D

#5 Look for a Pattern 4.OA.B.4 Step 1

Count days each grandchild calls.
The number of multiples of $k$ in days $1$ through $365$ is $\lfloor 365/k \rfloor$ — floor-divide $365$ by the call interval.

$$|A| = \lfloor 365/3 \rfloor = 121, \quad |B| = \lfloor 365/4 \rfloor = 91, \quad |C| = \lfloor 365/5 \rfloor = 73$$

💡 Counting multiples of a number in a range is the same Grade 4 'factors and multiples' skill, just applied three times.

#12 Draw a Venn Diagram 6.NS.B.4 Step 2

Count days when TWO specific grandchildren both call.
Both call only on days divisible by both intervals — i.e., multiples of $\text{lcm}$.
Pairwise: $\text{lcm}(3,4) = 12$, $\text{lcm}(3,5) = 15$, $\text{lcm}(4,5) = 20$.

💡 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).

#12 Draw a Venn Diagram 6.NS.B.4 Step 3

Count days when ALL THREE grandchildren call.
That requires the day to be a multiple of $\text{lcm}(3,4,5) = 60$ — the very center of the three-circle Venn diagram.

$$|A \cap B \cap C| = \lfloor 365/60 \rfloor = 6$$

💡 The triple-overlap days are multiples of the LCM of all three intervals — the deepest center of the Venn diagram.

#12 Draw a Venn Diagram 4.OA.A.3 Step 4

Apply inclusion-exclusion to find the total number of days with at least one call.
We add the three circle totals, subtract the three pairwise overlaps (each was counted twice), then add back the triple overlap (subtracted one time too many).

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

💡 The +/-/+ pattern is just the Venn diagram bookkeeping: each region of the picture gets counted exactly once when you finish.

#16 Change Focus / Count the Complement 4.NBT.B.4 Step 5

Apply the Complement trick.
Total days in $2017$ is $365$; days with at least one call is $219$; days with NO call is the rest.

$$365 - 219 = 146 \;\Rightarrow\; \textbf{(D)}$$

💡 'No call' is the complement of 'at least one call' — subtract from the total. Matches choice (D).

[1] #5 4.OA.B.4 Count days each grandchild calls. The number of multiples of $k$ in days $1$ thr

[2] #12 6.NS.B.4 Count days when TWO specific grandchildren both call. Both call only on days div

[3] #12 6.NS.B.4 Count days when ALL THREE grandchildren call. That requires the day to be a mult

[4] #12 4.OA.A.3 Apply inclusion-exclusion to find the total number of days with at least one cal

[5] #16 4.NBT.B.4 Apply the Complement trick. Total days in $2017$ is $365$; days with at least on

Review

Reasonableness: 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: $365 \times 24/60 = 146$ exactly. The result matches and lands on choice (D), with $146$ comfortably inside the $78$–$152$ option spread.

Alternative: Tool #2 (Systematic List) on the $60$-day cycle: write days $1$–$60$, cross out every multiple of $3$, $4$, $5$, and count the survivors. You'll find $24$ no-call days per $60$-day block. Since $365 = 6 \times 60 + 5$ and the first $5$ days $\{1, 2, 3, 4, 5\}$ of each cycle have exactly $2$ no-call days ($1$ and $2$), the total is $6 \times 24 + 2 = 146$ — same answer (D).

CCSS standards used (min grade 6)

4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite (Counting the multiples of $3$, $4$, and $5$ inside the $365$-day range via floor-division $\lfloor 365/k \rfloor$ — the Grade 4 multiples skill.)
6.NS.B.4 Find greatest common factor and least common multiple of two numbers (Computing $\text{lcm}(3,4)=12$, $\text{lcm}(3,5)=15$, $\text{lcm}(4,5)=20$, and $\text{lcm}(3,4,5)=60$ to count the days in each pairwise and triple Venn overlap.)
4.OA.A.3 Solve multi-step word problems using four operations with whole numbers (Combining the seven counts with the inclusion-exclusion $+$/$-$/$+$ pattern: $285 - 72 + 6 = 219$.)
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers (Performing the final complement subtraction $365 - 219 = 146$ to get the no-call count.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 6)

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