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the product of z and the complex number 5-6i is a real number. find two possible nonzero values of z.

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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.

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