2005 AMC 8 Problem 18 — Solution | LIVE by Po-Shen Loh

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2005 AMC 8 Problem 18

Attempt Problem 18 of the 2005 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 8 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

18.

How many three-digit numbers are divisible by $13?$

$7$

$67$

$69$

$76$

$77$

Answer: C

Concepts:counting integers in a range multiple

Difficulty rating: 1260

Solution:

We want to find how many $k$ exist such that $13k$ is a three-digit number.

The smallest possible $k$ such that $13k \gt 99$ is $k = 8.$

The largest possible $k$ such that $13k \lt 1000$ is $k = 76.$

This tells us that $k$ can range from $8$ to $76,$ which gives us $69$ values for $k.$

Thus, C is the correct answer.

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