Prove $4|10^n \iff n>1$

Question

I am just wondering if it is true that $4|10^n \iff n>1$. I was thinking that it is because $2|10$ and $2\cdot2=4$ so $4|10^2$ but not $10$ so $n > 1$.

Answer 1

$$n>1\implies 4\mid2^n\,,\;\;\text{but then:}\;\;4\mid 10^n=(2\cdot5)^n=2^n\cdot5^n$$

Answer 2

Yes it is true.

It becomes evident if you write $10^n = 2^n \cdot 5^n$.

Now you have $4 = 2^2 \mid 10^n = 2^n \cdot 5^n$, hence $n \ge 2$

Note that your explanation is perfectly fine, I just used a more standard mathematical writing :)

Answer 3

It's a simple matter of factoring. $4 = 2^2$ and $10^n = 2^n \times 5^n$. In order for $2^m$ (or $5^m$, for that matter) to divide $10^n$ the following inequality must hold: $m \leq n$.