Gaussian Integer?

Number Theory Level 2

$z = a+bi$ is a Gaussian integer .
The absolute value of $z$ is also a Gaussian integer.

Which of the following could represent $a$ and $b$ ?

8, 5 None of these pairs 1, 1 3, 4 1, 2 4, 5 5, 6 All of these pairs

1 solution

Geoff Pilling
Jan 20, 2017

The absolute value of $z$ is given by:

$r = |z| = \sqrt{a^2 + b^2}$

In order for this to be a Gaussian integer, implies that $\sqrt{a^2 + b^2}$ is an integer.

The only pair for which this is true is $\boxed{(3,4)}$ , since $3^2 + 4^2 = 5^2$