document.write( "Question 1182915: Prove that there are infinitely many natural numbers n such that √(19n+9) is irrational.\r
\n" ); document.write( "\n" ); document.write( "Thank you in advance! \n" ); document.write( "
Algebra.Com's Answer #813045 by ikleyn(52864)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "Prove that there are infinitely many natural numbers n such that \"sqrt%2819n%2B9%29\" is irrational.
\n" ); document.write( "~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "                The proof is very simple.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "                I'd say,  it is  UNEXPETEDLY  simple.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "Consider any integer positive number  \" n \",  which has the last digit of 2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(In other words, consider any integer positive number n such that  n = 2 (mod 10) ).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "For example, such numbers \"n\" are 12, 22, 32, 42, 52 . . . and so on.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then the number  19n  has the last digit of  8,  and the number  (19n+9)  has the last digit of 7,  because  8 + 9 = 17 = 7 (mod 10).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    +-------------------------------------------------------------------------+\r\n" );
document.write( "    |   But  NO  ONE  such integer positive number is a perfect square  (!)   |\r\n" );
document.write( "    +-------------------------------------------------------------------------+\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(Notice that a square of an integer number may have the last digit only 0, 1, 4, 5, 6, 9, but may not have the last digit of 2, 3, 7, 8).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Therefore, all the numbers  (19n +9)  with n = 2 (mod 10) are not perfect squares and, THEREFORE, create/produce  IRRATIONAL numbers \"sqrt%2819n%2B9%29\".\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "PROVED.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "and Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );