AMC 10 · 2009 · #12

Grade 7 number-theory
Archetype Small-Domain Constrained Search └ Sub-archetype Factor / Divisibility Constraint
digit-decompositiondigit-sumlinear-diophantine convert-to-algebradigit-constraints ↑ Prerequisites: place-value
📏 Medium solution 💡 3 insights
Problem
A number equals six times the sum of its own digits. Count how many such numbers there are below one thousand.

Pick an answer.

(A)
0
(B)
1
(C)
2
(D)
4
(E)
12
How to solve
Strategy Convert to Algebra

The condition ties a number to its own digits, so the first move is to stop treating the number as one object and write it by place value. Then "six times the digit sum" becomes a single linear equation in three digits. Digits are trapped between 0 and 9, and that bound alone is strong enough to force the hundreds digit, after which the rest is pinned by a divisibility argument. The final move is deliberately not algebra: every candidate the equation produces has to be tested against the original sentence, because the equation also accepts one number that the word "positive" rules out.

1STEP 1

Write the number out of its digits

Writing the number out names it by its three digits.

N = 100a + 10b + c, S = a + b + c, a,b,c ∈ {0,1,…,9}
2STEP 2

One equation replaces the search

One equation then replaces searching every number.

100a + 10b + c = 6(a + b + c) ⟹ 94a + 4b = 5c
3STEP 3

The hundreds digit must be 0

Comparing sizes forces the hundreds digit to zero.

a ≥ 1 ⟹ 94a + 4b ≥ 94 ＞ 45 ≥ 5c (impossible), so a = 0
4STEP 4

Solve 4b = 5c in digits

The rest is a small divisibility puzzle in two digits.

4b = 5c = V, gcd(4,5) = 1 → 20 ∣ V, V ≤ 36 → V ∈ {0, 20} → (b,c) = (0,0) or (5,4)
5STEP 5

Test the survivors against the words

Only one survivor is a positive number, choice (A).

(b,c) = (0,0) → N = 0 (not positive); (b,c) = (5,4) → N = 54, 6(5+4) = 54
Answer
1
Two independent size checks agree with a count of 1. First, any qualifying number is at most 6 x 27 = 162, so the entire search lives inside the first 162 integers; a count of 12 would mean roughly one in every fourteen of them works, which the equation 94a + 4b = 5c flatly forbids, since it pins the tens and units digits to a single nonzero pair. Second, the count cannot be 0, because 54 is exhibited and verified directly: 5 + 4 = 9 and 6 x 9 = 54. Sweeping 1 through 162 by hand confirms 54 is the only hit, so the answer is 1, choice (B).
💡Key takeaway

When a rule links a number to its own digits, write the number as 100a + 10b + c so both sides speak digits, then check every survivor against the original words.

  • Write the number out of its digits
  • One equation replaces the search
  • The hundreds digit must be 0
  • Solve 4b = 5c in digits
  • Test the survivors against the words

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