stonecold wrote:

Here 2^(any power ≥ 11) is a factor of three.Hence using the rule => Multiple - multiple is always a multiple

We must subtract a multiple of 3

I.e we must subtract 3

Smash that C for me.

I hope i am not missing anything

Your statement above (in red) is NOT correct.

Important point, 2^number where number >11 can not be a FACTOR of 3 but will be a MULTIPLE of 3. 2^12 is definitely NOT a multiple of 3 as 2^12 will only have 2s in it.

Coming back to the question,

2^1 leaves a remainder of 2 when divided by 3

2^2 leaves a remainder of 1 when divided by 3

2^3 leaves a remainder of 2 when divided by 3

2^4 leaves a remainder of 1 when divided by 3... etc. and the cyclicity continues.

Thus, \(2^{526}\) will leave a remainder of 1 when divided by 3. Thus you must subtract 1 from \(2^{526}\) to make it divisible by 3.

A is thus the correct answer.

Hope this helps.