Number Theory T...

In the complex numbers, where $$i = \sqrt {-1}$$, the conjugate of any value $$a + bi$$ is $$a -ib$$. What is the result when you multiply $$2 + 7i$$ by its conjugate?

The imaginary number $$i$$ is defined such that $$i^2=-1$$. What is the value of $$(1 - i \sqrt {5}) ( 1 + i\sqrt {5})$$?

In the complex numbers, where $$i^{2} = -1$$, what is the value of $$5 + 6i$$ multiplied by $$3-  2i$$?

$$\left( \frac{1+cos\dfrac{\pi}{8}-i  sin \dfrac{\pi}{8}}{1+cos \dfrac{\pi}{8}+i  sin \dfrac{\pi}{8}}\right)^8 =$$

How many of the prime factors of $$30$$ are greater than $$2$$? 

If $$i = \sqrt {-1}$$, find the values of $$n$$ such that $$i^{n} + (i)^{n}$$ have a positive value.

Find $$(5 + 2i)(5 - 2i)$$

Find the number of prime numbers between 301 and 320?

Express  $$\dfrac {(-5-i)(-7+8i)}{(2-4i)}$$  in the form of a complex number $$a+bi$$.

Every composite number has _____________.