Binomial Theorem

Given the expression 32n+2−8n−9, we want to determine the maximum power of 2 that divides this expression for all n∈ℕ.
First, we can rewrite 32n+2 as 9n+1 :
9n+1−8n−9
Considering 9=1+8, we can apply the binomial expansion to (1+8)n+1 :
(1+8)n+1=‌n+1C0+‌n+1C1⋅8+‌n+1C2⋅82+‌n+1C3⋅83+...
Simplifying, we get:
1+8(n+1)+82n+1C2+‌n+1C3⋅83+...
Subtracting 8n+9 from this sequence, we have:
1+8n+8+82n+1C2+...−8n−9
Which simplifies to:
82(‌n+1C2+8n+1C3+...)
The expression 82 can be further evaluated as 26. Thus, the binomial expansion shows that after simplification, the given expression is a multiple of 26.
Therefore, the maximum value of p such that the expression is divisible by 2p is:
6