NumbersThe expression 2^6n – 4^2n , where n is a natural number is always divisible by

To determine if the expression 2^6n - 4^2n is always divisible by a natural number, we can break it down step by step.

Step 1: Simplify the expression. 2^6n can be written as (2^6)^n, which is equal to 64^n. Similarly, 4^2n can be written as (4^2)^n, which is equal to 16^n.

So, the expression becomes 64^n - 16^n.

Step 2: Factor out common terms. We can factor out 16^n from both terms: 64^n - 16^n = (4^3)^n - (4^2)^n = (4^3 - 4^2)^n

Step 3: Simplify the expression further. 4^3 - 4^2 = 64 - 16 = 48

So, the expression becomes 48^n.

Step 4: Conclusion. Since 48^n is a multiple of 48 for any natural number n, we can conclude that the expression 2^6n - 4^2n is always divisible by a natural number.