The expression 2^6n – 4^2n , where n is a natural number is always divisible byOptions15364818

To determine if the expression 2^6n - 4^2n is always divisible by the given options, we can simplify the expression and check for divisibility.

Step 1: Simplify the expression. Using the properties of exponents, we can rewrite the expression as (2^6)^n - (4^2)^n. Simplifying further, we get 64^n - 16^n.

Step 2: Check for divisibility. To determine if this expression is always divisible by the given options, we need to check if it is divisible by each option individually.

Option 1: 15 To check if the expression is divisible by 15, we need to see if it leaves a remainder of 0 when divided by 15. We can substitute n = 1 to evaluate the expression: 64^1 - 16^1 = 64 - 16 = 48. Since 48 is not divisible by 15, the expression is not divisible by 15.

Option 2: 36 Substituting n = 1: 64^1 - 16^1 = 64 - 16 = 48. Since 48 is not divisible by 36, the expression is not divisible by 36.

Option 3: 48 Substituting n = 1: 64^1 - 16^1 = 64 - 16 = 48. Since 48 is divisible by 48, the expression is divisible by 48.

Option 4: 18 Substituting n = 1: 64^1 - 16^1 = 64 - 16 = 48. Since 48 is not divisible by 18, the expression is not divisible by 18.

Therefore, the expression 2^6n - 4^2n is only divisible by the option 48.