Find the smallest positive integer N

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SUMMARY

The smallest positive integer N that meets the specified conditions is derived from the product of the first twelve prime numbers, which are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and 37. To satisfy the conditions of being a square, a cube, and an odd number, N must take the form of 2*3*5*7*11*13*17*19*23*29*31*37*(2K+1)^6. When K is set to 0, N equals approximately 7.420738135 x 10^12, resulting in a total of 13 digits. This conclusion is based on the mathematical properties of squares and cubes in relation to prime factorization.

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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.
 
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Beer induced query follows.
sfvdsc said:
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?
 
jonah said:
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.
 
Just a quick note, if N is odd then it can't be divisible by 2!

-Dan
 

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