# Zero divisors in Zp where p is prime

#### sarah77

1. Homework Statement

Find all zero divisors of the ring Z17

2. Homework Equations

Are there any zero divisors of the ring Z17?

3. 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?

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#### Dick

Science Advisor
Homework Helper
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

Science Advisor
Homework Helper
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

Science Advisor
Homework Helper
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?

Thank you!

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