How does this imply this (number theory)

[dr hannibal]

so I have
$$2^{1990}=(199k+2)^{10}$$
expanding I have.
$$2^{1990}=2^{10}+10.2^9. (199k)+\frac{10.9}{1.2} 2^8.(199k)^2+...+10.2. (199k)^9+(199K)^{10}$$-(1)

now its clear $$199|2^{1990}-2^{10}$$ since I can take 199 out of the RHS.

but the book seems to imply that the above equation(1) says $$10|2^{1990}-2^{10}$$ , but how?I can't see how the equation above says the $$10|2^{1990}-2^{10}$$ is true..

Thanks.

 

[jgens]

Clearly, 2¹⁹⁹⁰-2¹⁰ = 2¹⁰(2¹⁹⁸⁰-1). Therefore, to show that this number is divisible by 10, it suffices to show that 2¹⁹⁸⁰-1 is divisible by 5. You can prove this fact by showing that 5|2⁴ⁿ-1 (use induction).

 

[dr hannibal]

Clearly, 2¹⁹⁹⁰-2¹⁰ = 2¹⁰(2¹⁹⁸⁰-1). Therefore, to show that this number is divisible by 10, it suffices to show that 2¹⁹⁸⁰-1 is divisible by 5. You can prove this fact by showing that 5|2⁴ⁿ-1 (use induction).

thanks:)