document.write( "Question 752482: find the least 5 digit number which is exactly divisible by 20,25,30?
\n" ); document.write( "and
\n" ); document.write( "find the least number of five digits that is exactly divisible by 16,18,24,and30? \n" ); document.write( "

Algebra.Com's Answer #457801 by Edwin McCravy(20060) $\"\"$ $\"About$

You can put this solution on YOUR website!
find the least 5 digit number which is exactly divisible by 20,25,30?
\n" ); document.write( "

\r\n" );
document.write( "20 = 2*10 = 2*2*5\r\n" );
document.write( "25 = 5*5\r\n" );
document.write( "30 = 2*15 = 2*3*5\r\n" );
document.write( "\r\n" );
document.write( "First we decide the least number of 2's we must have:\r\n" );
document.write( "a.  20 contains two 2's\r\n" );
document.write( "b.  25 contains no 2's\r\n" );
document.write( "c.  30 contains one 2\r\n" );
document.write( "\r\n" );
document.write( "So to contain 20 as a factor, our number must contain two 2's as factors.\r\n" );
document.write( "\r\n" );
document.write( "Next we decide the least number of 3's we must have:\r\n" );
document.write( "a.  20 contains no 3's\r\n" );
document.write( "b.  25 contains no 3's\r\n" );
document.write( "c.  30 contains one 3\r\n" );
document.write( "\r\n" );
document.write( "So to contain 30 as a factor, our number must contain one 3 as a factor\r\n" );
document.write( "\r\n" );
document.write( "First we decide the least number of 5's we must have:\r\n" );
document.write( "a.  20 contains one 5\r\n" );
document.write( "b.  25 contains two 5's\r\n" );
document.write( "c.  30 contains one 5\r\n" );
document.write( "\r\n" );
document.write( "So to contain 20 as a factor, our number must contain two 2's as factors\r\n" );
document.write( "\r\n" );
document.write( "The smallest integer to contain all those as factors is 2*2*3*5*5 = 300.\r\n" );
document.write( "\r\n" );
document.write( "Trouble is, 300 is only a 3-digit number, and we want a 5-digit number.\r\n" );
document.write( "so we must multiply 300 by the smallest integer that will make it be at\r\n" );
document.write( "least 10000, which is the smallest 5-digit number.\r\n" );
document.write( "\r\n" );
document.write( "We divide 10000 by 300 and get 33.33333...\r\n" );
document.write( "\r\n" );
document.write( "So the smallest integer greater than that is 34.\r\n" );
document.write( "\r\n" );
document.write( "So our number = 300*34 or 10200.\r\n" );
document.write( "\r\n" );
document.write( "----------------------------------------------------------------  \r\n" );
document.write( "

\n" ); document.write( " find the least number of five digits that is exactly divisible by 16,18,24,and 30?
\n" ); document.write( "

\r\n" );
document.write( "16 = 2*8 = 2*2*4 = 2*2*2*2\r\n" );
document.write( "18 = 2*9 = 2*3*3\r\n" );
document.write( "24 = 2*12 = 2*2*6 = 2*2*2*3\r\n" );
document.write( "30 = 2*15 = 2*3*5\r\n" );
document.write( "\r\n" );
document.write( "First we decide the least number of 2's we must have:\r\n" );
document.write( "a.  16 contains four 2's\r\n" );
document.write( "b.  18 contains one 2\r\n" );
document.write( "c.  24 contains three 2's\r\n" );
document.write( "d.  30 contains one 2.  \r\n" );
document.write( "\r\n" );
document.write( "So to contain 16 as a factor, our number must contain four 2's as factors.\r\n" );
document.write( "\r\n" );
document.write( "Next we decide the least number of 3's we must have:\r\n" );
document.write( "a.  16 contains no 3's\r\n" );
document.write( "b.  18 contains two 3's\r\n" );
document.write( "c.  24 contains one 3\r\n" );
document.write( "d.  30 contains one 3.\r\n" );
document.write( "\r\n" );
document.write( "So to contain 18 as a factor, our number must contain two 3's as factors\r\n" );
document.write( "\r\n" );
document.write( "First we decide the least number of 5's we must have:\r\n" );
document.write( "a.  16 contains no 5's\r\n" );
document.write( "b.  18 contains no 5's\r\n" );
document.write( "c.  24 contains no 5's\r\n" );
document.write( "d.  30 contains one 5.\r\n" );
document.write( "\r\n" );
document.write( "So to contain 30 as a factor, our number must contain one 5 as a factor.\r\n" );
document.write( "\r\n" );
document.write( "The smallest integer to contain all those as factors is 2*2*2*2*3*3*5 = 720.\r\n" );
document.write( "\r\n" );
document.write( "Trouble is, 720 is only a 3-digit number, and we want a 5-digit number.\r\n" );
document.write( "so we must multiply 720 by the smallest integer that will make it be at\r\n" );
document.write( "least 10000, which is the smallest 5-digit number.\r\n" );
document.write( "\r\n" );
document.write( "We divide 10000 by 720 and get 13.88888...\r\n" );
document.write( "\r\n" );
document.write( "So the smallest integer greater than that is 14.\r\n" );
document.write( "\r\n" );
document.write( "So our number = 720*44 or 10080.\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );