<s>[INST] $1^n-3^n-6^n+8^n$ is divisible by $10$ Prove that $1^n-3^n-6^n+8^n$ is divisible by $10$ for all $n\in\mathbb{N}$
It is divisible by $2$ and $5$ if we rearrange it will it be enough 
$(1^n -3^n)$ and $(6^n -8^n)$ is divisible by $2$. 
And
$(1^n-6^n)$ and $(8^n-3^n)$ is divisible by $5$.
Hence $\gcd(2,5)$ is $1$ and it is divisible by $2\cdot5=10$.
Is it correct? 
 [/INST] Your proof is OK, however, for completeness, you must further show why the binomials are divisible by $2$ and $5$. 
If you are familiar with modular arithmetic, it is:
$$1^n-3^n-6^n+8^n\equiv 1-3-6+8\equiv 0 \pmod {10}.$$
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