2003 AMC 10A Problem 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?

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Solution

Both

d

and

e

are distinct members of

\{2,3,5,7\},

and

10d+e

must also be prime. Start with the largest possible tens digit. For

d=7,

the choices

e=5

and

e=2

give the composite numbers

75

and

72,

while

e=3

gives the prime

73.

This produces

n=7\cdot3\cdot73=1533.

For

d=5,

the choices

e=7

and

e=2

give the composite numbers

57

and

52,

while

e=3

gives only

5\cdot3\cdot53=795.

Every case with

d\le3

is at most

3\cdot7\cdot37=777.

Hence

1533

is the largest valid value. The sum of its digits is

1 + 5 + 3 + 3 = 12.

Thus, the correct answer is A.