AMC 10 · 2005 · #15

Grade 8 number-theory
Archetype Pair-Sum Invariant └ Sub-archetype Universe Total Minus Known
place-valuedigit-sumdivisibility-rulesdigit-decomposition convert-to-algebrasystematic-enumeration ↑ Prerequisites: place-valuedigit-decompositiondivisibility-rules
📏 Medium solution 💡 3 insights
Problem
Four two-digit numbers add to 221, and their eight digits are all different and none is zero. Exactly one nonzero digit is therefore missing. Decide which digit cannot appear.

Pick an answer.

(A)
1
(B)
2
(C)
3
(D)
4
(E)
5
How to solve
Strategy Introduce a Variable

Chasing the four actual numbers means searching thousands of cases. Instead, notice that the sum only cares about two totals: the sum of the four tens digits and the sum of the four units digits. Name those T and U. The digit condition gives a second equation about T+U, and combining the two leaves a single divisibility fact that pins the missing digit exactly. Then build one concrete example so the case is not just possible on paper.

1STEP 1

Name what is missing

Naming the missing digit makes the used digits add to 45 minus it.

1+2+…+9=45, (sum of the eight digits)=45-d
2STEP 2

Split the sum by place value

Splitting by place value gives a second, exact equation.

10T+U=221
3STEP 3

Combine the two facts

Combining leaves one clean relation between two unknowns.

10T+(45-d-T)=221 ⟹ 9T=176+d
4STEP 4

Only one digit fits

A divisibility restriction forces the digit to be 4.

9 ∣ 176+d, 176=9 · 19+5, 6 ≤ 5+d ≤ 14 ⟹ 5+d=9 ⟹ d=4
5STEP 5

Show it can actually happen

An explicit example shows it really happens, so the answer is 4, choice (D).

13+25+86+97=221, {1,3,2,5,8,6,9,7}={1,…,9}∖{4}
Answer
4
Check the example by hand: 13+25=38, 38+86=124, 124+97=221. Correct, and the digits 1,3,2,5,8,6,9,7 are eight different nonzero digits with 4 absent. Also check the other four choices are impossible, which the divisibility step already guarantees: if the missing digit were 1,2,3 or 5, then 176+d would be 177,178,179 or 181, none divisible by 9, so 9T=176+d would have no whole-number solution for T.
💡Key takeaway

When digits are shuffled around, track two totals instead of four numbers: what the tens digits add to and what the units digits add to.

  • Name what is missing
  • Split the sum by place value
  • Combine the two facts
  • Only one digit fits
  • Show it can actually happen

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