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?

Algebra ->  Finance -> 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?       Log On


   

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) About Me  (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:

2x%5E2+=+5y%5E5

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 x%5E2+=+%282%5Ea%29%285%5Eb%29, where a and b are even; and
and y%5E5+=+%282%5Ec%29%285%5Ed%29, where c and d are multiples of 5

Then

2%2A%282%5Ea%29%285%5Eb%29+=+5%282%5Ec%29%285%5Ed%29

%282%5E%28a%2B1%29%29%285%5Eb%29+=+%285%5E%28d%2B1%29%29%282%5Ec%29

From that we know

c+=+a%2B1 and b+=+d%2B1

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

x%5E2+=+%282%5E4%29%285%5E6%29 --> 2x%5E2+=+%282%5E5%29%285%5E6%29
y%5E5+=+%282%5E5%29%285%5E5%29 --> 5y%5E2+=+%282%5E5%29%285%5E6%29

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

%282%5E5%29%285%5E6%29+=+500000

500000+=+2%28250000%29+=+2%28500%5E2%29+=+2x%5E2
500000+=+5%28100000%29+=+5%2810%5E5%29+=+5y%5E5

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
If the number is n, then
n=2a%5E2+=+5b%5E5

2a%5E2=5b%5E5%0D%0A%0D%0A%7B%7B%7Ba%5E2+=+expr%285%2F2%29b%5E5



Rationalize the denominator on the right:



a=sqrt%2810b%5E5%29%2F2%5E%22%22

2a=sqrt%2810b%5E5%29

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%2810%2A10%5E5%29

2a=sqrt%2810%5E6%29

2a=10%5E3%29

2a=1000

a=500

So,

n=2a%5E2+=+2%2A500%5E2=2%2A250000+=+500000


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

Edwin