GATE Overflow Test Series | Mock GATE | Test 1 | Question: 5

Question

If $a$ is a positive odd number and $b$ is a negative even number, then which of the following must be a positive even number?

• A. $a -b$
• B. $ab$
• C. $-ab$
• D. $a +b$

Answer 1

The following are the parity properties of even and odd numbers:

• $\text{even} \pm \text{even} = \text{even}$
• $\text{odd} \pm \text{odd} = \text{even}$
• $\text{even} \pm  \text{odd} = \text{odd}$
• $\text{even}\times \text{even} = \text{even}$
• $\text{even}\times \text{odd} = \text{even}$
• $\text{odd} \times \text{odd} = \text{odd}$

Now, $a = \text{positive odd number} = 2k + 1,b = \text{negative even number} = -2k, \:\:\text{where}\: k\in \mathbf{N}$

Lets check the options one by one

• A. $a-b = 2k+1 - (-2k) = 4k+1\:\text{(positive odd number)}$
• B. $ab = (2k+1)(-2k) = (+\: \text{ve odd}) \times (-\:\text{ve  even}) = -\:\text{ve even (negative even number)}$
• C. $-ab = -(2k+1)(-2k) = (2k+1)(2k) = (+\:\text{ve odd}) \times (+\:\text{ve even}) = +\text{ve even (positive even number)}$
• D. $a+b = 2k+1-2k = 1\:\text{(positive odd number)}$

So, the correct answer is $(C).$