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AMC 8 · 2008 · #3

Grade 4 arithmetic
Archetype Modular Cycle Frequency Count
modular-arithmeticpattern-recognitiondivisibility-rules modular-arithmeticpattern-recognition ↑ Prerequisites: multi-digit-arithmeticpattern-recognition
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Problem

If February is a month that contains Friday the 13th13^{\text{th}}, what day of the week is February 1?

Pick an answer.

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

Understand

Restated: February has a Friday the $13^{\text{th}}$. What day of the week is February $1$?

Givens: February $13$ is a Friday; Days of the week repeat every $7$ days; Answer choices: (A) Sunday, (B) Monday, (C) Wednesday, (D) Thursday, (E) Saturday

Unknowns: The day of the week of February $1$

Understand

Restated: February has a Friday the $13^{\text{th}}$. What day of the week is February $1$?

Givens: February $13$ is a Friday; Days of the week repeat every $7$ days; Answer choices: (A) Sunday, (B) Monday, (C) Wednesday, (D) Thursday, (E) Saturday

Plan

Primary tool: #12 Use Parity or Modular Arithmetic

Secondary: #8 Work Backwards

Weekdays cycle every $7$ days, so the only thing that matters when stepping back from February $13$ to February $1$ is the remainder of $12$ divided by $7$. Tool #12 (Use Parity or Modular Arithmetic) names that idea: $12 \equiv 5 \pmod 7$. Tool #8 (Work Backwards) tells us where to start — we know February $13$ is a Friday, and we want February $1$, so we walk backwards in time. Together, the moves are: go back $7$ days (one full week, same weekday), then go back the remaining $5$ days.

Execute — Answer: A

#8 Work Backwards 4.OA.A.3 Step 1
  • Find how many days February $1$ is before February $13$.
  • Subtract the dates.
$$13 - 1 = 12 \text{ days}$$

💡 Working backwards from a known day means measuring the gap first, just like reading a number line from right to left.

#12 Use Parity or Modular Arithmetic 4.OA.B.4 Step 2
  • Break $12$ days into full weeks plus leftover days.
  • One full week ($7$ days) lands on the same weekday, so it can be skipped.
$$12 = 7 + 5 \;\Rightarrow\; 12 \equiv 5 \pmod 7$$

💡 Dividing by $7$ and keeping the remainder is the modular arithmetic move — the weekly cycle erases the $7$, only the $5$ leftover days change the weekday.

#12 Use Parity or Modular Arithmetic 3.OA.D.9 Step 3
  • Apply the $7$-day skip.
  • Go back exactly $7$ days from Friday February $13$ — that lands on Friday February $6$, still a Friday.
$$\text{Feb } 13 \text{ (Fri)} - 7 \text{ days} = \text{Feb } 6 \text{ (Fri)}$$

💡 The Grade 3 "identify a pattern" move: every $7$ steps, the weekday repeats.

#8 Work Backwards 4.OA.A.3 Step 4
  • Now walk back the $5$ leftover days from Friday February $6$ to February $1$.
  • Count weekdays backwards: Friday, Thursday, Wednesday, Tuesday, Monday, Sunday.
$\text{Fri} \to \text{Thu} \to \text{Wed} \to \text{Tue} \to \text{Mon} \to \text{Sun}$ ($5$ steps back)

💡 Five backward steps from Friday lands on Sunday — the last leftover days do the actual weekday shift.

#12 Use Parity or Modular Arithmetic 4.OA.A.3 Step 5
  • Read off the answer.
  • February $1$ is a Sunday.
$$\text{February } 1 = \text{Sunday} \;\Rightarrow\; \textbf{(A)}$$

💡 The mod-$7$ shortcut plus a short backward count gives the weekday directly.

[1] #8 4.OA.A.3 Find how many days February $1$ is before February $13$. Subtract the dates.
[2] #12 4.OA.B.4 Break $12$ days into full weeks plus leftover days. One full week ($7$ days) lan
[3] #12 3.OA.D.9 Apply the $7$-day skip. Go back exactly $7$ days from Friday February $13$ — tha
[4] #8 4.OA.A.3 Now walk back the $5$ leftover days from Friday February $6$ to February $1$. Co
[5] #12 4.OA.A.3 Read off the answer. February $1$ is a Sunday.

Review

Reasonableness: Check forward instead of backward. If February $1$ is Sunday, then Sunday $+ 7$ days $=$ Sunday February $8$, and Sunday $+ 12$ days $=$ five weekdays after Sunday February $8$, which lands on Friday February $13$. That matches the problem, so February $1$ being Sunday is consistent.

Alternative: Tool #2 (Make a Systematic List): write the Fridays of February in order — $13, 6,$ and (going back another week) $-1$. So one week before February $6$ would be "February $-1$," meaning January $30$ is also a Friday. Counting forward from a Friday on "day $-1$": day $0$ is Saturday, day $1$ is Sunday. So February $1$ is a Sunday, giving (A).

CCSS standards used (min grade 4)

  • 3.OA.D.9 Identify arithmetic patterns and explain them using properties of operations (Recognizing that the weekday repeats every $7$ days, so adding or subtracting a multiple of $7$ does not change the weekday.)
  • 4.OA.A.3 Solve multistep word problems using the four operations, including problems with remainders (Setting up the gap of $12$ days from February $1$ to February $13$ and stepping backwards from a known Friday to find the unknown weekday.)
  • 4.OA.B.4 Find factors and multiples of a whole number (used here as division-with-remainder) (Splitting $12 = 7 + 5$ — one full week plus $5$ leftover days — which is the same as computing $12 \bmod 7 = 5$.)

⭐ Whole weeks are free moves — only the leftover days after dividing by $7$ change the weekday. That mod-$7$ shortcut turns a calendar puzzle into a one-line count.

⭐ Whole weeks are free moves — only the leftover days after dividing by $7$ change the weekday. That mod-$7$ shortcut turns a calendar puzzle into a one-line count.

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