AMC 10 · 2016 · #6
Grade 4 arithmeticPick an answer.
AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The question asks for a minimum, so this is Tool #14 (Extreme Principle): pin down how small the answer can possibly be, then show that small value is actually reachable. Two parts. First, find a hard floor: the two hundreds digits are different and each at least 1, so the sum S can't be too small. That floor lets Tool #3 (Eliminate Possibilities) knock out the digit sums 1, 2, and 3. Second, Tool #6 (Guess and Check) builds one concrete pair of numbers whose sum has digit sum 4, proving 4 is attainable. A bound plus a matching example pins the minimum exactly.
How small can S be?
Each hundreds digit is at least 1 and the two differ, so they sum to at least 1 + 2 = 3, forcing S ≥ 300.
Two different non-zero hundreds digits are at smallest a 1 and a 2, which already drags the total up to the 300s.
4.NBT.A.2Evaluate Finite DifferencesRule out digit sums 1, 2, 3
Since S ≥ 300, the only digit sum below 4 needs S = 300, but that repeats 0 in a column — banned. So 1, 2, 3 are out.
To make every digit of S tiny you would need zeros stacked on zeros, but the all-different rule blocks reusing 0 twice.
2.NBT.A.1Eliminate PossibilitiesBuild a sum with digit sum 4
104 + 296 = 400 uses all six digits differently, and 400 has digit sum 4 — reachable, so the minimum is 4. Answer (B).
Carries let the tens and units columns roll over to 0, so only the hundreds digit is left to count.
4.NBT.B.4Guess And CheckTo make a sum's digits small, use carries to roll columns over to zero — but two different non-zero leading digits keep the sum stuck in the 300s, so 4 is as low as it goes.
- How small can S be?
- Rule out digit sums 1, 2, 3
- Build a sum with digit sum 4