# Thread: Even numbers and odd numbers..

1. ## Even numbers and odd numbers..

If a,b,c and d are four positive real numbers such that the sum of a,b and c is even and the sum of b,c and d is odd, then a^2 - d^2 is neccesarily

2. ## Re: Even numbers and odd numbers..

Originally Posted by kohila
If a,b,c and d are four positive real numbers such that the sum of a,b and c is even and the sum of b,c and d is odd, then a^2 - d^2 is neccesarily
I think that you mean "a,b,c and d are four positive integers" otherwise no conclusion is possible.
If that is the case then you can show $\displaystyle a-d=2K-1$ which is odd and $\displaystyle a+d=2J+1-2b-2c$ which is also odd.

But $\displaystyle a^2-b^2=(a-b)(a+b)$ SO?

3. ## Re: Even numbers and odd numbers..

Hello, kohila!

I have a primitive solution with brute-force Listing.

$\displaystyle \text{If }a,b,c,d\text{ are four positive integers such that }a+b+c\text{ is even}$
$\displaystyle \text{and }b+c+d\text{ odd, then }a^2 - d^2\text{ is neccesarily }\_\_\_.$

Let $\displaystyle \begin{Bmatrix} e &=& \text{even} \\ o &=& \text{odd}\end{Bmatrix}$

Since $\displaystyle a+b+c$ is even, either (1) all are even or (2) exactly one is even.

Hence, there are four cases to consider:
. . (1) $\displaystyle a,b,c$ are even.
. . (2) $\displaystyle a$ even, $\displaystyle b,c$ odd.
. . (3) $\displaystyle b$ even, $\displaystyle a,c$ odd.
. . (4) $\displaystyle c$ even, $\displaystyle a,b$ odd.

$\displaystyle (1)\;[a\;b\;c\;d] \,=\,[e\;e\;e\;\_\,]$
. . .Since $\displaystyle b+c+d$ is odd,$\displaystyle d$ must be odd.
. . .Hence: $\displaystyle a$ is even, $\displaystyle d$ is odd.

$\displaystyle (2)\;[a\;b\;c\;d] \,=\,[e\;o\;o\;\_\,]$
. . .Since $\displaystyle b+c+d$ is odd, $\displaystyle d$ must be odd.
. . .Hence, $\displaystyle a$ is even, $\displaystyle d$ is odd.

$\displaystyle (3)\;[a\;b\;c\;d] \,=\,[o\;e\;o\;\_\,]$
. . .Since $\displaystyle b+c+d$ is odd, $\displaystyle d$ must be even.
. . .Hence, $\displaystyle a$ is odd, $\displaystyle d$ is even.

$\displaystyle (4)\;[a\;b\;c\;d] \,=\,[o\;o\;e\;\_\,]$
. . .Since $\displaystyle b+c+d$ is odd, $\displaystyle d$ must be even.
. . .Hence, $\displaystyle a$ is odd, $\displaystyle d$ is even.

In all cases, $\displaystyle a$ and $\displaystyle d$ have opposite parity.
Hence, their squares will have opposite parity.

The difference between two numbers of opposite parity is odd.

4. ## Re: Even numbers and odd numbers..

Originally Posted by kohila
If a,b,c and d are four positive real numbers such that the sum of a,b and c is even and the sum of b,c and d is odd, then a^2 - d^2 is neccesarily

Necessarily...odd? Also, a,b,c,d should be integers, otherwise we cannot claim anything.

We have
$\displaystyle a + b + c \equiv 0 (\mod 2)$
$\displaystyle b + c + d \equiv 1 (\mod 2)$

If we subtract the second equation from the first, we get $\displaystyle a-d \equiv 1 (\mod 2)$, therefore a-d is odd. This implies that a+d is also odd, since a+d = (a-d) + 2d. Therefore $\displaystyle (a-d)(a+d) = a^2 - d^2$ must be odd.

5. ## Re: Even numbers and odd numbers..

Thanks... I can get it from your explanation.

6. ## Re: Even numbers and odd numbers..

Much thanks for ur deep core conceptual clarification...