Zero divisors in Zp where p is prime

[sarah77]

Homework Statement

Find all zero divisors of the ring Z17

Homework Equations

Are there any zero divisors of the ring Z17?

The Attempt at a Solution

I multiplied 17*17=289...that is only divisible by 17, so I do not think there are any zero divisors...am I missing something?

 

[Dick]

No, there aren't any zero divisors in Z_17. But I'm not sure that 17*17=289 which is only divisible by 17 is a clear statement of the reason why not. If a*b is divisible by 17 then can a and b both not be divisible by 17? Why not?

 

[sarah77]

Both a and b have to be in Z17, so if a*b does not give 0 in Z17, it is not a zero divider, right?

 

[Dick]

Both a and b have to be in Z17, so if a*b does not give 0 in Z17, it is not a zero divider, right?

Sure, that's the definition. Do you think this might have anything to do with 17 being a prime number?

 

[sarah77]

Yes, but I wanted to explain it using a and b

 

[Dick]

Yes, but I wanted to explain it using a and b

If a*b=0 mod 17 then they are zero divisors. That means a*b is divisible by 17. Is that possible without a or b being divisible by 17?

 

[sarah77]

Thank you!