Competition · AMC preparation · step 4 of 4
AMC 8 · 2008 · #3
Grade 4 arithmeticPick an answer.
AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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 ≡ 5 (mod 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.
Count the days between
Subtract the dates: February 1 sits 12 days before February 13.
Working backwards from a known day means measuring the gap first, just like reading a number line from right to left.
4.OA.A.3Analyze The UnitsSplit into weeks and days
One full week returns to the same weekday, so 12 days leaves 5 leftover days.
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.
Stepping back 12 days from Friday the 13th lands on the same weekday as stepping back only 5 days.
▸ Why?
The 12 days split into one full week of 7 days plus 5 leftover days, and the full week lands back on the same weekday, so only those 5 leftover days move the weekday.
▸ Why?
12 days break with no gaps or overlaps into 7 days and 5 days, so stepping back 12 days is stepping back 7 days and then 5 more.
▸ Why?
The weekdays run in a fixed loop of length 7, so stepping a whole 7 days returns to the very same weekday.
Skip back one week
Skip one whole week: from Friday the 13th, 7 days back is Friday February 6.
The Grade 3 "identify a pattern" move: every 7 steps, the weekday repeats.
3.OA.D.9Draw A Venn DiagramWalk back five days
Count the 5 leftover days back from Friday: Thu, Wed, Tue, Mon, Sunday.
Five backward steps from Friday lands on Sunday — the last leftover days do the actual weekday shift.
4.OA.A.3Analyze The UnitsRead off the day
So February 1 is a Sunday → (A).
The mod-7 shortcut plus a short backward count gives the weekday directly.
4.OA.A.3Draw A Venn DiagramWhole 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.
- Count the days between
- Split into weeks and days
- Skip back one week
- Walk back five days
- Read off the day
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