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.2Extreme PrincipleRule 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.
Carries let the tens and units columns roll over to zero, so only the leading digit is left to count.
▸ Why?
A number is its digits sitting in fixed places, so a carry moves a full unit into the next place.
▸ Why?
A column rolls to zero exactly when its sum is a full multiple of ten, which the last place records.
To 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