AMC 8 · 2005 · #18

Grade 4 number-theory

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

multiplesdivisibility-rulesinterval-arithmetic identify-subproblems ↑ Prerequisites: multiplesdivisibility-rules

📏 Short solution 💡 2 insights

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Problem

Count how many three-digit numbers (the integers from 100 to 999) are divisible by 13.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

Counting three-digit multiples of 13 directly is awkward — the multiples (104, 117, 130, …) are not the numbers we want to count. Tool #9 (Easier Related Problem) replaces the task with something much simpler: each three-digit multiple has the form 13k, so we only need to count the values of k that make 13k a three-digit number. That turns the question into "how many integers k satisfy 100 ≤ 13k ≤ 999?" — a plain integer-counting problem. Tool #6 (Guess and Check) is the right way to find the two endpoints of k: try a few values near $\frac{100}{13}$ and $\frac{999}{13}$ until 13k first lands in, and last lands inside, the three-digit window. Once the endpoints are pinned down, the count is just (last - first) + 1.

1STEP 1

Every three-digit multiple of 13 is 13k, so count the integers k with 100 ≤ 13k ≤ 999 instead.

100 ≤ 13k ≤ 999, k ∈ {1, 2, 3, …}

2STEP 2

Test k = 8: 13 × 8 = 104 is the first three-digit multiple, so the smallest k is 8.

13 × 7 = 91 ＜ 100, 13 × 8 = 104 ≥ 100 → k_min = 8

3STEP 3

13 × 76 = 988 fits but 13 × 77 = 1001 has four digits, so the largest k is 76.

13 × 77 = 1001 ＞ 999, 13 × 76 = 988 ≤ 999 → k_max = 76

4STEP 4

The valid k run 8, 9, …, 76, so the count is (76 - 8) + 1 = 69.

(76 - 8) + 1 = 68 + 1 = 69 → (C)

Answer

A quick estimate confirms the count. There are 900 three-digit numbers (100 through 999), and roughly 1 in every 13 of them is a multiple of 13. That gives

\frac{900}{13}

≈ 69.2, which matches the exact count of 69. The answer choices (A) 7 and (D) 76 are red herrings — 7 is the largest k that is too small, and 76 is the largest valid k, not the count itself. Choices (B) 67 and (E) 77 are off-by-one or off-by-two mistakes from the (last - first) + 1 step. Only (C) 69 survives.

💡Key takeaway

Instead of counting three-digit multiples of 13, count the multipliers k. The smallest is 8 (since 13 × 8 = 104), the largest is 76 (since 13 × 76 = 988), and the integers from 8 to 76 number (76 - 8) + 1 = 69.

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