2018 AMC 12B Problem 15

Below is the professionally curated solution for Problem 15 of the 2018 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2018 AMC 12B solutions, or check the answer key.

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Concepts:digits divisibility multiplication principle

Difficulty rating: 1930

15.

How many $3$ -digit positive odd multiples of $3$ do not include the digit $3?$

$96$

$97$

$98$

$102$

$120$

Solution:

Write the number as $\overline{abc}.$ The hundreds digit $a$ has $8$ choices ( $1,2,4,5,6,7,8,9$ ), and the units digit $c$ has $4$ choices ( $1,5,7,9$ ).

The tens digit $b$ may be any of $\{0,1,2,4,5,6,7,8,9\}.$ These split into three residue classes mod $3$ of equal size $\{0,6,9\},\{1,4,7\},\{2,5,8\},$ so exactly $3$ choices of $b$ make $a+b+c$ divisible by $3.$

The count is $8\cdot4\cdot3=96.$

Thus, the correct answer is A.

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