Algebraic Proof

Prove algebraically that the sum of the squares of any two odd numbers leaves a remainder of 2 when divided by 4.

Ok so lets say the odd numbers are, 2n+1 and 2n+3

then
(2n+1)² + (2n+3)²
= 8n² + 16n + 10
divide this by 2 because you can...
4n²+8n+5

now what? thanks to whoever replies
manps
Prove algebraically that the sum of the squares of any two odd numbers leaves a remainder of 2 when divided by 4.

Ok so lets say the odd numbers are, 2n+1 and 2n+3

then
(2n+1)² + (2n+3)²
= 8n² + 16n + 10
divide this by 2 because you can...
4n²+8n+5

let the odd numbers be 2n+1 and 2n+1
then...

(2n+1)² + (2n+1)²
= 2(4n² + 4n + 1)
= 8n² + 8n + 2

- remainder 2 at the end.
manps
Prove algebraically that the sum of the squares of any two odd numbers leaves a remainder of 2 when divided by 4.

Ok so lets say the odd numbers are, 2n+1 and 2n+3

then
(2n+1)² + (2n+3)²
= 8n² + 16n + 10
divide this by 2 because you can...
4n²+8n+5

now what? thanks to whoever replies

use two different odd numbers not consecutive...call odd number one 2n+1 and number two 2m+1
You have to account for it being ANY two odd numbers. So, A = (2a+1) and B = (2b+1)

So, A^2 + B^2 = 4a^2 +4a^ +1 + 4b^2 + 4b + 1

= 4(a^2 + a + b^2 + b) + 2
Dekota
let the odd numbers be 2n+1 and 2n+1
then...

(2n+1)² + (2n+1)²
= 2(4n² + 4n + 1)
= 8n² + 8n + 2

- remainder 2 at the end.

you havent divided by 4
--------------
Da Bachtopus
You have to account for it being ANY two odd numbers. So, A = (2a+1) and B = (2b+1)

So, A^2 + B^2 = 4a^2 +4a^ +1 + 4b^2 + 4b + 1

= 4(a^2 + a + b^2 + b) + 2

ok but you havent divided by 4
manps
ok but you havent divided by 4
If you show that a number is 4k+2 you've shown it is 2 more than a multiple of 4, and therefore if you divded 4k+2 you've "k remainder 2", which is what is required.
2n+1 2n-1

(2n+1)^2 + (2n-1)^2

expand-
[(4n^2 +2n+2n+1)] + [(4n^2 -2n-2n-1)]
then I collected my like terms
which gave me
8n^2 then I divided it by 4
which gave me 2n^2 and the 2 before the n shows that dividing it by 4 would leave a remainder of 2
Original post by jellybean636
...