2023 AMC 12B Problem 23

Below is the professionally curated solution for Problem 23 of the 2023 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 12B solutions, or check the answer key.

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Concepts:prime factorization systematic listing

Difficulty rating: 2270

23.

When $n$ standard six-sided dice are rolled, the product of the numbers rolled can be any of $936$ possible values. What is $n?$

$11$

$6$

$8$

$10$

$9$

Solution:

Each die contributes an exponent vector in the primes $2,3,5$ (face $1\to(0,0,0),$ $2\to(1,0,0),$ $3\to(0,1,0),$ $4\to(2,0,0),$ $5\to(0,0,1),$ $6\to(1,1,0)$ ), and a product is determined by the sum of these vectors. Counting the distinct attainable sums for $n=1,2,3,\ldots$ gives $6,18,40,75,126,196,288,405,550,726,936,$ so $n=11.$

Thus, the correct answer is A.

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