Question 881095
{{{highlight(2)}}}
The prime factorization of {{{50}}} is
{{{50=2*5*5=2*5^2}}}.
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*5^2}}} , {{{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*50=2*(2*5^2)=2^2*5^2}}} has even exponents,
so {{{100=2^2*5^2=(2*5)^2=10^2}}} is a perfect square,
and the factor we needed to multiply times {{{50}}} to get that perfect square was {{{highlight(2)}}} .
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.