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Let 
. Then 
. By convention, every non-zero integer 
divides 0, so 
.
Suppose this relation holds for
, i.e. 
. We then hope to show it must also hold for 
.
You have
We assumed that, and it's clear that 
because 
is a multiple of 3. This means the remainder upon divides 
must be 0, and therefore the relation holds for . This proves the statement.
Suppose this relation holds for
You have
We assumed that
Option D: is the expression equivalent to
Explanation:
Option A:
The expression can be written as
Applying exponent rule, we get,
Thus, the expression is not equivalent to the expression
Hence, Option A is not the correct answer.
Option B:
The expression can be written as
Applying exponent rule, we get,
Thus, the expression is not equivalent to the expression
Hence, Option B is not the correct answer.
Option C:
The expression can be written as
Applying exponent rule, we get,
Thus, the expression is not equivalent to the expression
Hence, Option C is not the correct answer.
Option D:
The expression can be written as
Applying exponent rule, we get,
Thus, the expression is equivalent to the expression
Hence, Option D is the correct answer.