Question

Given that $a,b$ are odd and $c,d$ are even. Then which of the following is true:
A. ${{a}^{2}}-{{b}^{2}}+{{c}^{2}}-{{d}^{2}}$ is always divisible by $4$
B. $abc+bcd+cda+dac$ is always divisible by $4$
C. ${{a}^{4}}+{{b}^{4}}+{{c}^{3}}+{{d}^{3}}+{{c}^{2}}b+{{a}^{2}}b$ is always odd
D. $a+2b+3c+4d$ is odd

Answer 1

Hint: First write the definition of odd and even numbers. Then, we will pick some examples what are even and odd numbers from the sets $\left\{ 2k:k\in \mathbb{Z} \right\}$and $\left\{ 2k+1:k\in \mathbb{Z} \right\}$ respectively and then we will check the options and get the answer.

Complete step by step answer:
Let’s understand what are odd and even numbers,
The numbers which are divisible by $2$ are even numbers. Even numbers leave $0$ as a remainder when divided by $2$. Even numbers have $0,2,4,6\text{ or }8$ as their unit digit. For example: $2,4,16,246,80$ etc. are even numbers. The sets of even numbers are expressed as $\left\{ 2k:k\in \mathbb{Z} \right\}$ .
Odd numbers are the numbers which are not completely divisible by $2$. The odd numbers leave $1$ as a remainder when divided by $2$. They have $1,3,5,7\text{ or 9}$ as their unit digit. $11,123,179,95$ etc. are odd numbers. The sets of odd number are expressed as $\left\{ 2k+1:k\in \mathbb{Z} \right\}$
Let’s see the steps to check for odd and even numbers, first, we will divide the number by $2$ and then we will check the remainder of the remainder is $0$ it is an even integer else if the remainder is $1$, it is an odd number.
Now, it is given that $a,b$ are odd and $c,d$ are even. Now let’s assume $a=1$, $b=3$ that means odd numbers and then $c=2$ and $d=4$ ,
Now we will see that in option D , $a+2b+3c+4d=1+\left( 2\times 3 \right)+\left( 3\times 2 \right)+\left( 4\times 4 \right)=1+6+6+16=29$ which means it is an odd number.
Hence, option D is correct.

Note:
The other three options are always not correct for different values $a,b,c\text{ and }d$ . Note that decimals are not even or odd numbers because they are not whole numbers. For example, you can’t say that the fraction $\dfrac{1}{3}$ is odd by the fact that a denominator is an odd number or $12.34$ as an even as its last digit is even. Only integers can be even, or odd, meaning decimals and fractions cannot be even or odd. Zero, however, is an integer and is divisible by two, so it is even.