1991 AMC 12 Problem 17

Question

A positive integer

N

is a palindrome if the integer obtained by reversing the sequence of digits of

N

is equal to

N.

The year

1991

is the only year in the current century with the following two properties:

(a) It is a palindrome.

(b) It factors as a product of a

2

-digit prime palindrome and a

3

-digit prime palindrome.

How many years in the millennium between

1000

and

2000

(including the year

1991

) have properties (a) and (b)?

1

2

3

4

5

Every four-digit palindrome from

1000

to

2000

has the form

1dd1

Factor

1dd1

as

11(91+10d)

and test when the three-digit factor is prime

The palindromes in the interval are

1dd1,

where

d

is a digit. Algebraically,

\begin{aligned} 1dd1&=1001+110d\\ &=11(91+10d). \end{aligned}

The two-digit prime palindrome must therefore be

11,

and

91+10d

must be a three-digit prime palindrome. For

d=0,1,\ldots,9,

the prime values occur for

d=1,4,6,9,

giving

101,131,151,181.

Thus there are

4

years.

Thus the correct answer is D.

Problem 17 in other years