Question

Find all the zero divisors of 15.

Answer 1

In order to find all the zero divisors in $\mathbb{Z}_{15}$, we will solve the equation $xy=0$ (mod 15) in integers.

Since 15 divides xy, then 3 and 5 divide xy.

Since 3 is a prime number, at least one of x, y must be divisible by 3. We may assume without loss of generality that x is divisible by 3.

Since 5 is a prime number, at least one of x, y must be divisible by 5. If x is divisible by 5, then it is divisible by 15 (since 3 and 5 are coprime), i.e. x=0 mod 15. Hence, we do not obtain non-trivial solution and any zero divisors.

If y is divisible by 5, then x=3k, y=5n, $xy=3k\cdot 5n=15nm=0$ (mod 15). Therefore, all the zero divisors modulo 15 are multiples of 3 and multiples of 5. In $\mathbb{Z}_{15}$ they are 3,6,9,12 and 5,10.

By the definition, 0 is not counted as a zero divisor.

Answer: 3,6,9,12 and 5,10.