# 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,
\$\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 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