Problem 18 · 2005 AMC 8
Easy
Number Theory
count-multiples
How many three-digit numbers are divisible by 13?
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Answer: C — 69.
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Hint 1 of 2
Every multiple of 13 is 13·k. Instead of hunting the multiples, hunt the multipliers k that keep 13k in the 100–999 range — they form a tidy run of consecutive whole numbers.
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Hint 2 of 2
Find the first k with 13k ≥ 100 and the last with 13k ≤ 999. Then count the integers between, inclusive.
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Approach: count the multipliers, not the multiples
- Write each multiple as 13k. Smallest 3-digit one: 100 ÷ 13 ≈ 7.7, so k = 8 gives 13·8 = 104. Largest: 999 ÷ 13 ≈ 76.8, so k = 76 gives 13·76 = 988.
- So k runs through 8, 9, …, 76 — a block of consecutive integers. Count them: 76 − 8 + 1 = 69.
- The two things people botch: remember the +1 (counting endpoints, not gaps — that's why it's 69, not 68), and translate the problem into counting k's rather than the multiples themselves. This 'count the index' trick works for any 'how many multiples in a range' question.
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