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AMC 8 · 2002 · #5

Grade 4 number-theory
Archetype Modular Cycle Frequency Count
modular-arithmeticdivisibility-rulespattern-recognition modular-arithmeticpattern-recognition ↑ Prerequisites: multi-digit-arithmeticdivisibility-rules
📏 Short solution 💡 2 insights
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Problem

Carlos Montado was born on Saturday, November 9, 2002. On what day of the week will Carlos be 706 days old?

Pick an answer.

(A)
Monday
(B)
Wednesday
(C)
Friday
(D)
Saturday
(E)
Sunday
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Toolkit + CCSS Solution

Understand

Restated: Carlos was born on Saturday, November 9, 2002. Day $0$ of his life is that Saturday. On what day of the week will day $706$ of his life fall?

Givens: Day $0$ (birth day) is a Saturday; We want the day of the week on day $706$; Answer choices: (A) Monday, (B) Wednesday, (C) Friday, (D) Saturday, (E) Sunday

Unknowns: The day of the week $706$ days after a Saturday

Understand

Restated: Carlos was born on Saturday, November 9, 2002. Day $0$ of his life is that Saturday. On what day of the week will day $706$ of his life fall?

Givens: Day $0$ (birth day) is a Saturday; We want the day of the week on day $706$; Answer choices: (A) Monday, (B) Wednesday, (C) Friday, (D) Saturday, (E) Sunday

Plan

Primary tool: #5 Look for a Pattern

Secondary: #9 Solve an Easier Related Problem

The week is a length-$7$ repetition cycle, which is the signature trigger for Tool #5 (Look for a Pattern). Once we see the cycle, the only thing that controls the answer is the remainder when $706$ is divided by $7$ — everything else is a multiple of $7$ that just lands back on Saturday. Tool #9 (Solve an Easier Related Problem) supports this: we replace $706$ with the nearby multiple of $7$ that is easy to handle ($700 = 7 \times 100$), solve that toy version in one line, and then add the small leftover.

Execute — Answer: C

#5 Look for a Pattern 4.OA.C.5 Step 1
  • Name the cycle.
  • The days of the week repeat in a loop of length $7$.
  • So adding any multiple of $7$ to a Saturday gives a Saturday again.
  • This means we only care about $706 \bmod 7$ — the remainder after taking out as many full weeks as possible.
$$706 = 7q + r, \quad 0 \le r < 7$$

💡 Spotting the length-$7$ cycle turns a big day count into a small leftover count.

#9 Solve an Easier Related Problem 4.OA.A.3 Step 2
  • Solve an easier version first.
  • Use the nearest multiple of $7$ below $706$, which is $700 = 7 \times 100$.
  • Day $700$ is exactly $100$ full weeks after birth, so day $700$ is the same weekday as day $0$ — a Saturday.
$$700 = 7 \times 100 \;\Rightarrow\; \text{day } 700 \text{ is Saturday}$$

💡 Replacing $706$ with the easier number $700$ makes the leftover tiny: just $6$ extra days.

#5 Look for a Pattern 4.OA.C.5 Step 3
  • Walk forward the leftover $6$ days from Saturday.
  • After day $700$ (Saturday): day $701$ Sunday, day $702$ Monday, day $703$ Tuesday, day $704$ Wednesday, day $705$ Thursday, day $706$ Friday.
$$\text{Sat} \to \text{Sun} \to \text{Mon} \to \text{Tue} \to \text{Wed} \to \text{Thu} \to \text{Fri}$$

💡 Six small steps along the weekday cycle, starting from the anchor Saturday.

#5 Look for a Pattern 4.OA.A.3 Step 4
  • Read off the answer.
  • Day $706$ is Friday, which matches choice (C).
$$706 \bmod 7 = 6 \;\Rightarrow\; \text{Saturday} + 6 \text{ days} = \text{Friday} \;\Rightarrow\; \textbf{(C)}$$

💡 The remainder $6$ is all the information the cycle needs to pick the weekday.

[1] #5 4.OA.C.5 Name the cycle. The days of the week repeat in a loop of length $7$. So adding a
[2] #9 4.OA.A.3 Solve an easier version first. Use the nearest multiple of $7$ below $706$, whic
[3] #5 4.OA.C.5 Walk forward the leftover $6$ days from Saturday. After day $700$ (Saturday): da
[4] #5 4.OA.A.3 Read off the answer. Day $706$ is Friday, which matches choice (C).

Review

Reasonableness: Check by going the other direction: $707 = 7 \times 101$, so day $707$ is exactly $101$ full weeks after birth — another Saturday. One day before Saturday is Friday, so day $706$ is Friday. Same answer (C). Either anchor ($700$ or $707$) gives the same Friday, which is the sign that the cycle is being used correctly.

Alternative: Tool #2 (Make a Systematic List) on a very small scale: list the seven possibilities for $706 \bmod 7$. Compute $706 \div 7 = 100$ remainder $6$. So shift Saturday forward by $6$ days using the weekday list Sat, Sun, Mon, Tue, Wed, Thu, Fri — landing on Friday, choice (C).

CCSS standards used (min grade 4)

  • 4.OA.C.5 Generate a number or shape pattern that follows a given rule; identify apparent features of the pattern that were not explicit in the rule itself (Recognizing the weekday cycle of length $7$ and using the rule "adding $7$ returns to the same weekday" to reduce day $706$ to a short walk from day $700$.)
  • 4.OA.A.3 Solve multistep word problems posed with whole numbers, including problems in which remainders must be interpreted (Dividing $706$ by $7$ and interpreting the remainder $6$ as the number of weekday steps forward from Saturday.)

⭐ Weekdays repeat every $7$ days, so for any "how many days later" question, divide by $7$ and use only the remainder. $706 \div 7$ leaves $6$, so jump $6$ days forward from Saturday and land on Friday — answer (C).

⭐ Weekdays repeat every $7$ days, so for any "how many days later" question, divide by $7$ and use only the remainder. $706 \div 7$ leaves $6$, so jump $6$ days forward from Saturday and land on Friday — answer (C).

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