2005 AMC 10A Problem 16

Below is the professionally curated solution for Problem 16 of the 2005 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 10A solutions, or check the answer key.

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Concepts:digits place value units digit

Difficulty rating: 1510

16.

The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is $6.$ How many two-digit numbers have this property?

$5$

$7$

$9$

$10$

$19$

Solution:

If the number is $10a + b,$ then $(10a + b) - (a + b) = 9a.$ The units digit of $9a$ is $6$ only when $a = 4,$ since $9 \cdot 4 = 36.$ The digit $b$ can then be anything from $0$ to $9,$ giving the ten numbers $40$ through $49.$

Thus, the correct answer is D.

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