SOLUTION: Find the least number by which 50 is multiplied / divided to make product a perfect square?

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Question 881095: Find the least number by which 50 is multiplied / divided to make product a perfect square?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
highlight%282%29
The prime factorization of 50 is
50=2%2A5%2A5=2%2A5%5E2.
Whatever the prime factorization of a positive integer,
when you square that integer,
in the prime factorization of the square, all exponents will be even numbers.
So if a number is a perfect square, all exponents in its prime factorization will be even numbers.
Conversely, if all the exponents in the prime factorization of a number are even, then the number is a perfect square.
The prime factorization of 50=2%2A5%5E2 , 5 has the even exponent 2 ,
but 2 has the odd exponent 1 .
So we know that 50 is not a perfect square.
To get a different number, the least integer that we can multiply times 50 is 2 .
Then 100=2%2A50=2%2A%282%2A5%5E2%29=2%5E2%2A5%5E2 has even exponents,
so 100=2%5E2%2A5%5E2=%282%2A5%29%5E2=10%5E2 is a perfect square,
and the factor we needed to multiply times 50 to get that perfect square was highlight%282%29 .
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.