document.write( "Question 881095: Find the least number by which 50 is multiplied / divided to make product a perfect square? \n" ); document.write( "

Algebra.Com's Answer #532050 by KMST(5328) $\"\"$ $\"About$

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$\"highlight%282%29\"$
\n" ); document.write( "The prime factorization of $\"50\"$ is
\n" ); document.write( " $\"50=2%2A5%2A5=2%2A5%5E2\"$ .
\n" ); document.write( "Whatever the prime factorization of a positive integer,
\n" ); document.write( "when you square that integer,
\n" ); document.write( "in the prime factorization of the square, all exponents will be even numbers.
\n" ); document.write( "So if a number is a perfect square, all exponents in its prime factorization will be even numbers.
\n" ); document.write( "Conversely, if all the exponents in the prime factorization of a number are even, then the number is a perfect square.
\n" ); document.write( "The prime factorization of $\"50=2%2A5%5E2\"$ , $\"5\"$ has the even exponent $\"2\"$ ,
\n" ); document.write( "but $\"2\"$ has the odd exponent $\"1\"$ .
\n" ); document.write( "So we know that $\"50\"$ is not a perfect square.
\n" ); document.write( "To get a different number, the least integer that we can multiply times $\"50\"$ is $\"2\"$ .
\n" ); document.write( "Then $\"100=2%2A50=2%2A%282%2A5%5E2%29=2%5E2%2A5%5E2\"$ has even exponents,
\n" ); document.write( "so $\"100=2%5E2%2A5%5E2=%282%2A5%29%5E2=10%5E2\"$ is a perfect square,
\n" ); document.write( "and the factor we needed to multiply times $\"50\"$ to get that perfect square was $\"highlight%282%29\"$ .
\n" ); document.write( "There is no smaller positive integer we could use, because multiplying $\"50\"$ times $\"1\"$ would just give us $\"50\"$ , and we know that $\"50\"$ is not a perfect square. \n" ); document.write( "

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