sarah77 said: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?
sarah77 said:Yes, but I wanted to explain it using a and b
Zero divisors in Zp where p is prime refer to elements in the ring of integers modulo p that, when multiplied by any other element, result in a product of 0. In other words, these elements have no multiplicative inverse in the ring.
To identify zero divisors in Zp, you can check if the given element is relatively prime to p. If the greatest common divisor of the element and p is not equal to 1, then the element is a zero divisor. Another way to identify zero divisors is to check if the element is congruent to 0 modulo p.
Studying zero divisors in Zp is important because it helps us understand the structure of the ring of integers modulo p. It also has applications in cryptography and coding theory, where zero divisors can be used to generate error-correcting codes.
The number of zero divisors in Zp is equal to the number of integers less than p that are not relatively prime to p. In other words, the number of zero divisors in Zp is equal to p - 1 minus the number of prime factors of p - 1.
No, a prime number cannot be a zero divisor in Zp. This is because a prime number is only divisible by 1 and itself, and in the ring of integers modulo p, every element has a unique multiplicative inverse except for 0. Therefore, a prime number cannot be multiplied by any other element to result in a product of 0.