To find the values of z that make the product with the complex number 5-6i a real number, we need to consider the imaginary part of the product.
The product of z and 5-6i can be written as:
z * (5 - 6i)
Expanding this expression, we get:
5z - 6zi
For the product to be a real number, the imaginary part (-6zi) must be equal to zero. This means that the coefficient of the imaginary unit i, which is -6z, must be zero.
Setting -6z = 0, we find:
z = 0
So, one possible nonzero value of z is 0.
However, since we are looking for nonzero values of z, we need to find another value that satisfies the condition.
Let's consider the equation for the imaginary part:
-6z = 0
Dividing both sides of the equation by -6, we have:
z = 0/(-6)
z = 0
Again, we find z = 0, which is not a nonzero value.
Therefore, there are no other nonzero values of z that make the product with the complex number 5-6i a real number. The only value that satisfies the condition is z = 0.