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

Please explain clearly...(Crying)

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- October 22nd 2012, 05:10 AMkohilaEven 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

Please explain clearly...(Crying) - October 22nd 2012, 06:56 AMPlatoRe: Even numbers and odd numbers..
- October 22nd 2012, 04:07 PMSorobanRe: Even numbers and odd numbers..
Hello, kohila!

I have a primitive solution with brute-force Listing.

Quote:

Let

Since is even, either (1) all are even or (2) exactly one is even.

Hence, there are four cases to consider:

. . (1) are even.

. . (2) even, odd.

. . (3) even, odd.

. . (4) even, odd.

. . .Since is odd, must be odd.

. . .Hence: is even, is odd.

. . .Since is odd, must be odd.

. . .Hence, is even, is odd.

. . .Since is odd, must be even.

. . .Hence, is odd, is even.

. . .Since is odd, must be even.

. . .Hence, is odd, is even.

In all cases, and have opposite parity.

Hence, their squares will have opposite parity.

The difference between two numbers of opposite parity is.*odd*

- October 22nd 2012, 06:56 PMrichard1234Re: Even numbers and odd numbers..
- October 22nd 2012, 07:31 PMkohilaRe: Even numbers and odd numbers..
**Thanks... I can get it from your explanation.** - October 22nd 2012, 07:45 PMkohilaRe: Even numbers and odd numbers..
**Much thanks for ur deep core conceptual clarification...**