AMC 10 · 2014 · #16

Grade 6 number-theory
Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search
digit-sumpattern-recognitionmulti-digit-arithmetic pattern-recognitionconvert-to-algebra ↑ Prerequisites: digit-sum
📏 Medium solution 💡 2 insights
Problem
A repdigit is multiplied by a single digit and the product's digits add to a target. Find its length.

Pick an answer.

(A)
901
(B)
911
(C)
919
(D)
991
(E)
999
How to solve
Strategy Look for a Pattern

A number with k eights is far too big to multiply directly for the size of k we expect. So compute a few small cases and hunt for a pattern (Tool #5): the products 8·88, 8·888, and so on. Once the digit shape of the product is clear, name the count of eights with a variable k (Tool #4) to write the digit sum as a formula in k, then turn "digit sum =1000" into a simple equation and solve it (Tool #13).

1STEP 1

Multiply the first few cases

Small cases show a clear pattern.

8·88=704, 8·888=7104, 8·8888=71104, 8·88888=711104
2STEP 2

Read off the digit pattern

The product has a fixed head and tail.

8·88…8_k=711…1_k-204
3STEP 3

Write the digit sum in terms of k

So the digit sum is linear in the length.

7+(k-2)+0+4=k+9
4STEP 4

Solve for k

Solving gives 991, choice (D).

k+9=1000 → k=991 → (D)
Answer
991
Test the formula on a case we computed: with k=3 the product is 7104, whose digits sum to 7+1+0+4=12, and the formula gives k+9=3+9=12. It matches, so trusting k+9=1000 gives k=991. Choice (D) also sits sensibly just below 999, which fits a digit sum barely over k.
💡Key takeaway

Multiply a few small cases, catch the digit pattern, and the giant problem shrinks to k+9=1000.

  • Multiply the first few cases
  • Read off the digit pattern
  • Write the digit sum in terms of k
  • Solve for k

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