This confuses me!

It would be equivalent to find the largest $n$ such that $2^n|1025^{1024}-1$ .

We factor the expression as

$1025^{1024}-1=(1025-1)(1025+1)(1025^2+1)(1025^4+1)(1025^8+1)...(1025^{512}+1)$

The greatest power of $2$ which divides the first factor is $2^{10}$ . Since $1025=2^{10}+1$ , the terms of the expansions within each of the remaining ten factors only contain powers of two, the smallest being $2$ itself. Thus, the largest power of two which divides $1025^{1024}-1$ is $2^{10+10}=2^{20}.$

$\boxed{n=20}$

I would have written a solution using the Exponent Lifting Lemma, but somebody else has already done so. 😏

Note that $1025=1024+1$ , then the expression becomes

${ (1024+1) }^{ 1024 }-1=\\= \sum _{ k=0 }^{ 1023 }{ \left( \begin{matrix} 1024 \\ k \end{matrix} \right) { 1024 }^{ 1024-k } } =\\ ={ 1024 }^{ 1024 }+\left( \begin{matrix} 1024 \\ 1 \end{matrix} \right) { 1024 }^{ 1023 }+...+\left( \begin{matrix} 1024 \\ 1022 \end{matrix} \right) { 1024 }^{ 2 }+\left( \begin{matrix} 1024 \\ 1023 \end{matrix} \right) 1024$

Note that every term, except the last one, is divisible by ${ 2 }^{ 21 }$ , as the powers of $1024={ 2 }^{ 10}$ are increasing from right to left. Thus, the smallest $n$ such that $2^{n+1}\nmid 1025^{1024}-1$ is $20$ , because ${ 2 }^{ 21 }\nmid { 1024 }^{ 2 }$ , which is the last term.

Let $V_2(x)$ denote the largest power of 2 which divides $x$ .

Since $\gcd(1025,2) =1$ and $\gcd(1,2) = 1$ , we can use Lifting the Exponent Lemma here.

By lifting the exponent lemma,

$V_2(1025^{1024}-1) = V_2(1025^{1024}-1^{1024}) = V_2(1025-1)+V_2(1025+1)+V_2(1024)-1$

Since $V_2(1024) = 10$ , $V_2(1026) = 1$ , $V_2(1025-1)+V_2(1025+1)+V_2(1024)-1 = 20$

Therefore, $2^{20} \mid 1025^{1024}-1$ , $2^{21} \nmid 1025^{1024}-1$

So, $n+1 = 21$ , $n = \boxed{20}$

1025^1 = 1 (mod 2^10) , 1025^2 = 1 (mod 2^11) , 1025^4 = 1 (mod 2^12) , 1025^8 = 1 (mod 2^13) , .
There is a pattern look at the powers of 1025 and the powers of 2
The pattern is 2^0 = 2^10 , 2^1 = 2^11 , . 2^2 = 2^12 ,

1025^1024 =1 (mod 2^20) ,

that means (1025^1024)-1 is divisible by 2^20
2^21 is our answer and so the value of n is 20