I'm working on a problem where I need to determine the conditions under which $2^n - 46$ is divisible by 4, where $n$ is a non-negative integer. I understand that for any power of 2 greater than $2^2$, the result is always divisible by 4. However, when subtracting 46 from $2^n$, I'm unsure how to systematically approach or solve for $n$ to ensure the result remains divisible by 4.
I've considered direct computation for small values of $n$ and observed patterns, but I'm looking for a more general method or a mathematical insight that could help solve this more efficiently or elegantly. Specifically, I'm interested in any theorems, properties, or techniques that could be applied to this problem.
Could anyone provide guidance on how to approach this problem or point me towards relevant mathematical concepts or methods that could simplify determining the divisibility of $2^n - 46$ by 4?
Thank you in advance for any assistance!
If $n=0$ then $2^n-46=-45$ isn’t divisible by $4$.
If $n=1$ then $2^n-46=-44$ is divisible by $4$.
If $n\ge 2$ then $2^n-46$ isn’t divisible by $4$ as a difference of a number divisible by $4$ and a number not divisible by $4$.
You could write $2^n-46 = 4(2^{n-2}-12)+2$. Number $n-2$ is non-negative, hence $2^{n-2}-12$ is integer. Also, $2<4$. That is the division with remainder formula. The remainder is $2$, not $0$.