Competition · AMC preparation · step 4 of 4
AMC 8 · 2017 · #7
Grade 4 number-theoryPick an answer.
AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Instead of jumping into algebra, we use Tool #9 (Easier Problem) by working with the concrete example 247247 the problem hands us, plus one or two more abcabc numbers. Tool #5 (Pattern) helps us notice that every abcabc number is exactly abc × 1001 — once we see that pattern, the question reduces to "which choices divide 1001?". Finally Tool #3 (Eliminate Possibilities) checks the five choices against 1001 to pick the winner. We deliberately avoid Tool #13 (Algebra) — the place-value pattern is easier to discover by playing with one example than by setting up variables first.
Divide the given example
Start with the given example: divide 247247 by its repeated block 247 to get 1001.
Dividing a 6-digit number by a 3-digit number is a Grade 4 long-division skill — no algebra needed.
4.NBT.B.6Solve An Easier Related ProblemTest another six-digit number
Confirm it's no fluke: 315315 ÷ 315 also gives 1001.
Computing two cases and seeing the same quotient 1001 is the Grade 4 way of confirming a number pattern before trusting it.
4.OA.C.5Look For A PatternExplain 1001 by place value
By place value the front abc is worth 1000 copies, so abcabc = abc × 1001.
Reading abcabc by place value — the first abc is worth 1000 times the second abc — is exactly the Grade 4 multi-digit place-value standard.
Any number that repeats its three-digit block, like abcabc, is exactly that block times 1001.
▸ Why?
The six-digit number splits with no gaps or overlaps into the top three-digit block and the bottom three-digit block, so its value is the top part plus the bottom part.
▸ Why?
The top block sits three places higher than the bottom block, so it is worth 1000 times the bottom block, which makes the number abc × 1000 + abc.
▸ Why?
Each place to the left is worth ten times the place before it, so shifting a block three places up multiplies its value by ten three times, which is 1000.
▸ Why?
abc × 1000 plus abc × 1 is abc×(1000+1)=abc × 1001, because the shared block can be pulled out of both terms.
Factor 1001 into primes
Any factor of 1001 divides Z, so factor it: 1001 = 7 × 11 × 13.
Finding factor pairs of a whole number (here, 1001) is the Grade 4 factor-pair / prime-or-composite standard.
4.OA.B.4Look For A PatternCheck the choices against 1001
Test each choice against 1001: only 11 divides it (1001 ÷ 11 = 91); the rest leave remainders.
Testing each choice as a divisor of 1001 is the same Grade 4 factor-pair check applied to a multiple-choice elimination — only the true factor stays standing.
4.OA.B.4Eliminate PossibilitiesThis AMC 8 problem only needs Grade 4 place value and factor pairs you already know — once you see that abcabc = abc × 1001, it's just asking which choice divides 1001!
- Divide the given example
- Test another six-digit number
- Explain 1001 by place value
- Factor 1001 into primes
- Check the choices against 1001
A parent dashboard for the family lives at sensimlab.com.