Competition · AMC preparation · step 4 of 4
AMC 8 · 2002 · #5
Grade 4 number-theoryPick an answer.
AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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 × 100), solve that toy version in one line, and then add the small leftover.
Set up the 7-day cycle
Name the cycle: weekdays repeat every 7 days, so multiples of 7 return to Saturday — only 706 mod 7 matters.
Spotting the length-7 cycle turns a big day count into a small leftover count.
4.OA.C.5Look For A PatternSolve day 700 first
Solve an easier version: 700 = 7 × 100 is exactly 100 full weeks after birth, so day 700 is again a Saturday.
Replacing 706 with the easier number 700 makes the leftover tiny: just 6 extra days.
Day 700 of Carlos's life falls on the very same weekday as the day he was born.
▸ Why?
The gap of 700 days is exactly 100 whole weeks, with no extra days spilling over.
▸ Why?
700 = 7 × 100, which is 100 equal groups of 7 days, and 7 days is one week — so 700 days packs perfectly into 100 weeks.
▸ Why?
Finishing any whole week brings you back to the weekday you started on, so 100 finished weeks land back on the starting weekday.
▸ Why?
The weekdays run in a fixed repeating loop of length 7, so every 7 steps the loop returns to the position it began at.
Step forward the leftover days
Walk the leftover 6 days from Saturday: Sun, Mon, Tue, Wed, Thu, and day 706 lands on Friday.
Six small steps along the weekday cycle, starting from the anchor Saturday.
4.OA.C.5Look For A PatternRead off the weekday
Read off the answer: day 706 is Friday — choice (C).
The remainder 6 is all the information the cycle needs to pick the weekday.
4.OA.A.3Look For A PatternWeekdays repeat every 7 days, so for any "how many days later" question, divide by 7 and use only the remainder. 706 ÷ 7 leaves 6, so jump 6 days forward from Saturday and land on Friday — answer (C).
- Set up the 7-day cycle
- Solve day 700 first
- Step forward the leftover days
- Read off the weekday
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