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

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- May 7th 2006, 09:50 AMmathlgIntegers
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 AMThePerfectHackerQuote:

Originally Posted by**mathlg**

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

I believe the theorem goes that,

$\displaystyle ax=b$ has a solution in $\displaystyle \mathbb{Z}_n$ when, $\displaystyle \gcd(b,n)=1$.

For your problem you need to find all the integers relatively prime to $\displaystyle 20$ and less. This is called the 'phi-function'. Thus, you need to find the smalles $\displaystyle n>20$ 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 AMmathlg
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 PMThePerfectHackerQuote:

Originally Posted by**mathlg**