Integers

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May 7th 2006, 09:50 AM
mathlg

Integers

Find an integer n > 20 so that the equation 3x = b always have a unigue solution in Zn regardless of the value b

Not to sure what Zn means
Can the value of b be anything?

Please help
May 7th 2006, 10:07 AM
ThePerfectHacker

Quote:

Originally Posted by mathlg

Find an integer n > 20 so that the equation 3x = b always have a unigue solution in Zn regardless of the value b

Not to sure what Zn means
Can the value of b be anything?

Please help

By,
$\mathbb{Z}_n$ we mean the group of positive integers added modulo $n$ .

I believe the theorem goes that,
$ax=b$ has a solution in $\mathbb{Z}_n$ when, $\gcd(b,n)=1$ .
For your problem you need to find all the integers relatively prime to $20$ and less. This is called the 'phi-function'. Thus, you need to find the smalles $n>20$ such as all the previous integers are relatively prime to it. Meaning a prime number, the smallest after 20 is 23.
May 7th 2006, 10:20 AM
mathlg

So you are saying that since n > 20 the next smallest prime number is 23. That would be the integer im looking for in the problem? I get confused when it says regardless of the value b.
May 7th 2006, 01:13 PM
ThePerfectHacker

Quote:

Originally Posted by mathlg

So you are saying that since n > 20 the next smallest prime number is 23. That would be the integer im looking for in the problem? I get confused when it says regardless of the value b.

Yes