document.write( "Question 550071: How many positive integers n are there such that nē - 14n + 24 is a prime number? \n" ); document.write( "
Algebra.Com's Answer #358388 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "A prime number is an integer greater than 1 \r\n" );
document.write( "that has no other factors beside itself and 1\r\n" );
document.write( "\r\n" );
document.write( "nē - 14n + 24\r\n" );
document.write( "\r\n" );
document.write( "can be factored as\r\n" );
document.write( "\r\n" );
document.write( "(n - 2)(n - 12)\r\n" );
document.write( "\r\n" );
document.write( "Therefore unless one of the factors (n - 12) or (n - 2) is 1, and the other\r\n" );
document.write( "a prime number, (n - 2)(n - 12) will not be prime.\r\n" );
document.write( "\r\n" );
document.write( "If (n - 2) equals 1 then\r\n" );
document.write( "\r\n" );
document.write( "    n - 2 = 1\r\n" );
document.write( "        n = 3\r\n" );
document.write( "\r\n" );
document.write( "But (3 - 2)(3 - 12) = 1(-9) = -9\r\n" );
document.write( "\r\n" );
document.write( "But that is a negative number so (n - 2) can't be the factor that is 1.\r\n" );
document.write( "\r\n" );
document.write( "If (n - 12) equals 1 then\r\n" );
document.write( "\r\n" );
document.write( "    n - 12 = 1\r\n" );
document.write( "        n = 13\r\n" );
document.write( "\r\n" );
document.write( "And (13 - 2)(13 - 12) = 11(1) = 11 which IS prime.\r\n" );
document.write( "\r\n" );
document.write( "So 13 is the only number n can be such \r\n" );
document.write( "that nē - 14n + 24 is prime.\r\n" );
document.write( "\r\n" );
document.write( "The questios is how many positive integers is that? \r\n" );
document.write( "The answer is one, the number 13.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );