AMC 8 · 2025 · #13
Grade 4 number-theory
Pick an answer.
AMC 8 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Dividing 2, 4, 6, … by 7 produces a repeating cycle of remainders, so Tool #5 (Look for a Pattern) is the natural primary tool — compute the first several remainders, spot the cycle, then count efficiently. Tool #2 (Systematic List) supports this by listing the first seven remainders in order so the cycle is visible. Tool #3 (Eliminate Possibilities) is the multiple-choice closer: once we have the seven counts, only one of (A)-(E) matches and the rest can be crossed out.
The even numbers 2, 4, …, 50 are 2·1 through 2·25, so there are exactly 25 of them — the seven bars must sum to 25.
Pairing each even number with a counting number 1-25 is a Grade 4 multi-step word-problem move.
4.OA.A.3Make A Systematic ListListing the remainders of the first even numbers when divided by 7 exposes a length-7 cycle: 2, 4, 6, 1, 3, 5, 0.
Finding whole-number quotients and remainders is exactly the Grade 4 division-with-remainder skill.
4.NBT.B.6Make A Systematic List16 leaves remainder 2, restarting the cycle; adding 14 (= 2·7) never changes the remainder mod 7, so the 7-block repeats forever.
Spotting a repeating cycle in a generated number list is the Grade 4 pattern-rule standard in action.
4.OA.C.5Look For A PatternDividing 25 by 7 gives 25 = 7·3 + 4, so the cycle runs 3 full times (21 numbers) with 4 leftover entries.
Splitting 25 into 3 full groups of 7 plus 4 leftovers is the same Grade 4 division-with-remainder idea, used at the cycle level.
4.NBT.B.6Look For A PatternAfter 3 cycles every remainder sits at 3; the leftovers 44, 46, 48, 50 give remainders 2, 4, 6, 1, so bump those four bins by 1.
Combining "3 from each cycle" with "+1 for the leftovers" is a Grade 4 two-step word-problem combination.
4.OA.A.3Make A Systematic ListIn remainder order 0–6 the bar heights are 3, 4, 4, 3, 4, 3, 4, summing to 25 — matching the even-number count.
Reading bar heights off a frequency tally is the Grade 3 scaled-bar-graph skill.
3.MD.B.3Eliminate PossibilitiesOnly histogram (A) shows [3, 4, 4, 3, 4, 3, 4]; (B), (C), (D), (E) each differ in a bar, so the answer is (A).
Comparing our computed frequencies to each histogram is a Grade 3 bar-graph interpretation task.
3.MD.B.3Eliminate PossibilitiesThis AMC 8 problem only needs Grade 4 division-with-remainder and pattern-spotting you already know!
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Same archetype — closest grade level first.
