document.write( "Question 814037: If the product 19845k is a perfect cube, what is the smallest possible value of k if k is a positive integer? \n" ); document.write( "
Algebra.Com's Answer #490095 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "19845k = 3*3*3*3*5*7*7k = 34*51*72k\r\n" );
document.write( "\r\n" );
document.write( "by a value that will make it\r\n" );
document.write( "a perfect cube, we much choose a value that will cause\r\n" );
document.write( "every factor to have an exponent which is a multiple of 3.\r\n" );
document.write( "\r\n" );
document.write( "The 34 must be multiplied by 32 so it\r\n" );
document.write( "will become 36 \r\n" );
document.write( "\r\n" );
document.write( "The 51 must be multiplied by 52 so it\r\n" );
document.write( "will become 53\r\n" );
document.write( "\r\n" );
document.write( "The 72 must be multiplied by 71 so it\r\n" );
document.write( "will become 73\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So we must multiply 34*51*72 by\r\n" );
document.write( "32*52*71\r\n" );
document.write( "\r\n" );
document.write( "So k = 32*52*71 = 9*25*7 = 1575\r\n" );
document.write( "\r\n" );
document.write( "So the smallest possible positive integer value of k is 1575\r\n" );
document.write( "\r\n" );
document.write( "Checking:\r\n" );
document.write( "\r\n" );
document.write( "19845*1575 = 31255875 = 315³ \r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );