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smc

number theory senior

Problem

A positive integer is a palindrome if the integer obtained by reversing the sequence of digits of is equal to . The year 1991 is the only year in the current century with the following 2 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 millenium between 1000 and 2000 have properties (a) and (b)?
(A)
(B)
(C)
(D)
Solution
Notice that all four-digit palindromes are divisible by , so that is our two-digit prime. Because the other factor is a three-digit number, we are looking at palindromes between and , which also means that the last digit of the three-digit number is . Checking through the three-digit numbers , we find out that there are three-digit prime numbers, which when multiplied by , result in palindromes.
Final answer
D