Difference between revisions of "1991 AHSME Problems/Problem 17"
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Solution by e_power_pi_times_i | Solution by e_power_pi_times_i | ||
| − | Notice that all four-digit palindromes are divisible by <math>11</math>, so that is our two-digit prime. Because the other factor is a three-digit number, we are looking at palindromes between <math>1100</math> and <math>2000</math>, which also means that the last digit of the three-digit number is <math>1</math>. Checking through the three-digit numbers <math>101, 111, 121,\dots, 191</math>, we find out that <math>\boxed{\textbf{(D) 4}}</math> three-digit numbers, when multiplied by <math>11</math>, result in palindromes. | + | Notice that all four-digit palindromes are divisible by <math>11</math>, so that is our two-digit prime. Because the other factor is a three-digit number, we are looking at palindromes between <math>1100</math> and <math>2000</math>, which also means that the last digit of the three-digit number is <math>1</math>. Checking through the three-digit numbers <math>101, 111, 121,\dots, 191</math>, we find out that <math>\boxed{\textbf{(D) 4}}</math> three-digit prime numbers, which when multiplied by <math>11</math>, result in palindromes. |
== See also == | == See also == | ||
Revision as of 11:36, 13 December 2016
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)?
Solution
Solution by e_power_pi_times_i
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 
three-digit prime numbers, which when multiplied by , result in palindromes.
See also
| 1991 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
