Question 1133242
<pre>
If the number is n, then
{{{n=2a^2 = 5b^5}}}

{{{2a^2=5b^5

{{{a^2 = expr(5/2)b^5}}}

{{{matrix(2,3,"","","",a,""="",sqrt(expr(5/2)b^5))}}}

Rationalize the denominator on the right:

{{{a=sqrt(expr(5/2)b^5)=sqrt(expr(5/2)b^5*expr(2/2))=sqrt(10b^5/4^"")=sqrt(10b^5)/2^""}}}

{{{a=sqrt(10b^5)/2^""}}}

{{{2a=sqrt(10b^5)}}}

The left side is a positive integer, so the right side 
is also.  The smallest value of b that will cause the 
right side to be a positive integer is 10. 

{{{2a=sqrt(10*10^5)}}}

{{{2a=sqrt(10^6)}}}

{{{2a=10^3)}}}

{{{2a=1000}}}

{{{a=500}}}

So,

{{{n=2a^2 = 2*500^2=2*250000 = 500000}}}


So 500000 is the smallest positive integer which is 2 times 
the square of 500 and also 5 times 10<sup>5</sup>?

Edwin</pre>