2003 AMC 10A Problem 14

Below is the professionally curated solution for Problem 14 of the 2003 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 10A solutions, or check the answer key.

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Concepts:prime digits systematic listing

Difficulty rating: 1500

14.

Let $n$ be the largest integer that is the product of exactly $3$ distinct prime numbers, $d,$ $e,$ and $10d + e,$ where $d$ and $e$ are single digits. What is the sum of the digits of $n?$

$12$

$15$

$18$

$21$

$24$

Solution:

Both $d$ and $e$ are single-digit primes, and $10d + e$ must be prime. Testing the largest options, $75$ and $57$ are not prime.

Using $d = 7,$ $e = 3$ gives the prime $73,$ and $n = 7 \cdot 3 \cdot 73 = 1533.$

The sum of its digits is $1 + 5 + 3 + 3 = 12.$

Thus, the correct answer is A.

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