MHB Find the smallest positive integer N

[sfvdsc]

Find the smallest positive integer N that satisfies all of the following conditions:
• N is a square.
• N is a cube.
• N is an odd number.
• N is divisible by twelve prime numbers.
How many digits does this number N have?Please Explain your steps in detail.

 

[jonah1]

Beer induced query follows.

Find the smallest positive integer N that satisfies all of the following conditions:
• N is a square.
• N is a cube.
• N is an odd number.
• N is divisible by twelve prime numbers.
How many digits does this number N have?Please Explain your steps in detail.

What have you done so far?

 

[sfvdsc]

Beer induced query follows.

What have you done so far?

Dear Sir, I have been doing some rough, but I was unable to find the answer.

I have attached the rough.

So, in according to this if you can solve the problem. I will be very grateful to you.

The trouble is the divisibility by the first 12 prime numbers,
so it must be a multiple of 2*3*5*7*11*13*17*19*23*29*31*37

To be odd it must look like 2K+1

to be a square it must look like (2K+1)^2, and it must also be a cube
it must contain (2K+1)^6

so, it must have the form:
2*3*5*7*11*13*17*19*23*29*31*37(2K+1)^6
when K = 0, we get
2*3*5*7*11*13*17*19*23*29*31*37(1)^6
= 7.420738135... x 10^12
which would be 13 digits long

Please correct me if I am wrong!

I am really trying, please if anybody can solve this.

So, Please if you can solve this, I can use your answer as a reference and solve other related problems.

Please help me with my situation.

 

[topsquark]

Just a quick note, if N is odd then it can't be divisible by 2!

-Dan