Find the smallest positive integer N

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Discussion Overview

The discussion revolves around finding the smallest positive integer N that meets specific criteria: being a square, a cube, an odd number, and divisible by twelve prime numbers. Participants are exploring the implications of these conditions and seeking to determine the number of digits in N.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Post 1 introduces the problem and requests detailed steps for finding N.
  • Post 2 reiterates the problem and asks participants what they have done so far to solve it.
  • Post 3 discusses the challenges of meeting the divisibility condition by the first twelve prime numbers and proposes a form for N based on these conditions, suggesting that if K=0, N would be approximately 7.420738135... x 10^12, resulting in 13 digits.
  • Post 4 points out that if N is odd, it cannot be divisible by 2, which challenges the earlier assumptions made in the discussion.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are conflicting views regarding the implications of N being odd and its divisibility by 2. The discussion remains unresolved with multiple competing perspectives on how to approach the problem.

Contextual Notes

There are limitations regarding the assumptions made about the properties of N, particularly concerning its oddness and divisibility by 2, which may affect the validity of proposed forms for N.

sfvdsc
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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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