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\n" ); document.write( "Find perfect square factors of the given number and divide by those perfect squares; continue until what is left is NOT a perfect square. That will be the answer to the first question.
\n" ); document.write( "Note you only need to check for squares of prime numbers; if you have already found that the remaining number is not divisible by 2^2=4, then it won't be divisible by the square of any other even number.
\n" ); document.write( "The task is easy for me, because mental arithmetic is easy for me. For most people, a general strategy would be simply to check for divisibility by squares of the smallest prime numbers.
\n" ); document.write( "Inspection shows that 19404 is divisible by 4, because the last two digits are divisible by 4: 19404 = 4*4851.
\n" ); document.write( "The remaining number is not divisible by 2^2=4, so next check 3^2=9.
\n" ); document.write( "Inspection next shows the number is divisible by 9, because the sum of the digits is divisible by 9: 4851 = 9*539.
\n" ); document.write( "Those first two steps were easy if you know some basic divisibility rules. But it gets only a little bit harder from here if you follow the basic process of checking for divisibility by squares of prime numbers.
\n" ); document.write( "Clearly the number remaining is not divisible by 5^2=25, so try 7^2=49: 539 = 49*11.
\n" ); document.write( "So the original number is
\n" ); document.write( "19404 = 4*9*49*11
\n" ); document.write( "Now we can answer the first question: Dividing 19404 by 11 yields a perfect square.
\n" ); document.write( "From there the second question is easy. Dividing 19404 by 11 yields
\n" ); document.write( "4*9*49 = (2^2)(3^2)(7^2)
\n" ); document.write( "So the number whose square is the new number is 2*3*7 = 42. \n" ); document.write( "