SOLUTION: What is the smallest positive integer which is 2 times the square of some positive integer and also 5 times the fifth power of some other positive integer?

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Question 1133242: What is the smallest positive integer which is 2 times the square of some
positive integer and also 5 times the fifth power of some other positive integer?
Found 2 solutions by greenestamps, Edwin McCravy:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


We are looking for the smallest positive integer that is both 2 times the square of some positive integer and 5 times the 5th power of another positive integer:



There is a single known factor of 2 on the left and a single known factor of 5 on the right. That means both x and y must both contain factors of only 2 and/or 5. So

let , where a and b are even; and
and , where c and d are multiples of 5

Then





From that we know

and

For the smallest positive integer for which all those are true, we need to have c = d = 5; that makes a=4 and b=6.

And we finally have

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So the smallest positive integer that is both 2 times the square of one positive integer and 5 times the 5th power of another positive integer is
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
If the number is n, then






Rationalize the denominator on the right:







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. 











So,




So 500000 is the smallest positive integer which is 2 times 
the square of 500 and also 5 times 105?

Edwin
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