# 2003 AMC 10A Problems/Problem 14

## Problem

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 $n$? $\mathrm{(A) \ } 12\qquad \mathrm{(B) \ } 15\qquad \mathrm{(C) \ } 18\qquad \mathrm{(D) \ } 21\qquad \mathrm{(E) \ } 24$

## Solution 1

Since you want $n$ to be the largest number possible, you will want $d$ in $10d+e$ to be as large as possible. So $d = 7$.Then, $e$ cannot be $5$ because $10(7)+5 = 75$ which is not prime. So $e = 3$. $~~~$ $d \cdot e \cdot (10d+e) = 7 \cdot 3 \cdot 73 = 1533$. So, the sum of the digits of $n$ is $1+5+3+3=12 \Rightarrow \boxed{\mathrm{(A)}\ 12}$ ~ MathGenius_

## Solution 2

Since $d$ is a single digit prime number, the set of possible values of $d$ is $\{2,3,5,7\}$.

Since $e$ is a single digit prime number and is the units digit of the prime number $10d+e$, the set of possible values of $e$ is $\{3,7\}$.

Using these values for $d$ and $e$, the set of possible values of $10d+e$ is $\{23,27,33,37,53,57,73,77\}$

Out of this set, the prime values are $\{23,37,53,73\}$

Therefore the possible values of $n$ are: $2\cdot3\cdot23=138$ $3\cdot7\cdot37=777$ $5\cdot3\cdot53=795$ $7\cdot3\cdot73=1533$

The largest possible value of $n$ is $1533$.

So, the sum of the digits of $n$ is $1+5+3+3=12 \Rightarrow \boxed{\mathrm{(A)}\ 12}$

## Video Solution by WhyMath

~savannahsolver

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