Question

Use division algorithm to show that

Answer 1

What confuses me about this is that I think I am able to show that the square of any integer is in the form X∗k where x is any integer. For Example:

x=3q+0x=3q+1x=3q+2

I show 3k first

(3q)2=3(3q2)

where k=3q2 is this valid use of the division algorithm?

If it is then can I also say that int is in the form for example 10*k

for example

(3q)2=10∗(910q2)

where k=(910q2)

Why isn't this valid? Am I using the div algorithm incorrectly to show that any integer is the form 3k and 3k+1, if so how do I use it? Keep in mind I am teaching myself Number Theory and the only help I can get is from you guys in stackexchange