Question

Prove the following statements using either direct or contrapositive proof. If \(a \equiv 0(\bmod 4)\) or \(a \equiv 1(\bmod 4),\) then \(\left(\frac{a}{2}\right)\) is even.

Short answer

The statement 'If \(a \equiv 0(\bmod 4)\) or \(a \equiv 1(\bmod 4),\) then \(\left(\frac{a}{2}\right)\) is even' is only true in instances where \(a \equiv 0(\bmod 4)\). The latter condition doesn't suffice as dividing by 2 doesn't yield an even number.

Step-by-step solution

Understand the numerical condition

Firstly, understand that \(a \equiv 0 (\bmod 4)\) means that a is divisible by 4, and \(a \equiv 1(\bmod 4)\) means that a leaves a remainder of 1 when divided by 4. Both conditions imply that a is of the form \(4m\) or \(4m + 1\) respectively, where m is an integer.

Apply the condition and prove the resultant is even

Secondly, apply the conditions \(a = 4m\) or \(a = 4m + 1\). When \(a = 4m\), upon dividing by 2, we get \(a/2 = 2m\) which is an even number. Similarly, in the case where \(a = 4m + 1\), on dividing by 2, we get \(a/2 = 2m + 0.5\), which is not an integer and hence not even. Hence the statement is true only for the first condition, and false for the second.

Conclude

The statement is true only in cases where \(a \equiv 0 (\bmod 4)\), because upon division by 2, such a number is always even. The case of \(a \equiv 1(\bmod 4)\) doesn't satisfy the condition as dividing by 2 does not yield an even number.