Question

Prove or disprove:
For any integer n, the number n² + n + 1 is odd.

Answer 1

i think that's true

Answer 2

i think that to but i have to prove some how

Answer 3

5^2 + 5 +1
25 + 5 +1 = 31 (odd)

1^2 + 1 + 1
1 + 1 +1 = 3 (odd)

Answer 4

so we just proved it

Answer 5

break it up into cases

Case 1: If n is even,...

Case 2: If n is odd,...

Answer 6

i can't find any number that disproves it...

Answer 7

AriPotta you proved it works for those particular values, but you didn't prove it for all integers

Answer 8

2^2 + 2 + 1
4 + 2 + 1 = 7 (odd)

Answer 9

ok i'll try using cases

Answer 10

Case 1: n is an even integer

Let k be any integer, so 2k is always even.

Let n = 2k


n^2 + n + 1

(2k)^2 + (2k) + 1

4k^2 + 2k + 1

2(2k^2 + k) + 1

2q + 1 ... Let q = 2k^2 + k (you can easily show that q is always an integer)

So if n = 2k (ie if n is an even integer), then n^2 + n + 1 is equivalent to 2q + 1, which is an odd integer.

I'll let you do case 2

Answer 11

thank you!

Answer 12

An even + an odd is an odd . an odd + odd is an even number, and an even + even is an even number. If n is odd n^2 is odd so we have an odd + odd + 1 which is the same as odd + even so it will be odd. If n is an even n^2 is even so n^2 +n will be even. If you add 1 to an even number it is odd.

Aripotta has the right idea. Testing numbers works for odd and even in all cases.

Answer 13

np

Answer 14

thank you all of you!