# Thread: Number 1 Is Odd

1. ## Number 1 Is Odd

I know that 1 is an odd number but the reason is not clear to me. Why is 1 an odd number?

2. ## Re: Number 1 Is Odd

a number is odd if it is not divisible by 2

a number is odd if it can be written in the form $2k+1$ where $k$ is any integer

3. ## Re: Number 1 Is Odd Originally Posted by Cervesa a number is odd if it is not divisible by 2

a number is odd if it can be written in the form $2k+1$ where $k$ is any integer
Ok. Explain the 2k + 1 form.

4. ## Re: Number 1 Is Odd

Many questions in basic mathematics come down to the definitions!
What definition of "odd number" are you using?

I would say that an "odd number" is any integer that is not even. Or course, then I have to say what "even" means! An "even number" is an integer that is "evenly divisible" by 2: n is even if and only if n/2= k where k is an integer. That means that any even number is of the form n= 2k for some integer k.

So what about odd numbers? Since an odd number is not even, if n is odd, n/2 cannot be an integer. Dividing n by 2 must gives some integer, k, plus a remainder. We can write n/2= k+ j/2 where j is the non-zero remainder.

But the remainder when ever we divide one integer by another must be smaller than the divisor. When the divisor is 2, the only possible remainders are 0 and 1. Since j was to be "non-zero" it must be 1.

That is, if n is odd, n/2= k+ 1/2. Multiplying both sides by 2, n= 2k+ 1.

Any even number can be written in the form n= 2k for some integer k, any odd number can be written in the form n= 2k+ 1.

(If k is an integer, so is k+ 1. If we write k'= k+1, then k= k'- 1 so n= 2(k'- 1)+ 1= 2k'- 2+ 1= 2k'- 1. Any odd number can also be written as "n= 2k- 1" just not the same k as before.)

Another way of looking at this is that "even" and "odd" numbers alternate: given any even number, the number on either side, 1 more or 1 less, is odd.

5. ## Re: Number 1 Is Odd Originally Posted by harpazo Ok. Explain the 2k + 1 form.
1 = 2(0) + 1

zero is an integer

6. ## Re: Number 1 Is Odd Originally Posted by Cervesa 1 = 2(0) + 1

zero is an integer
Why equate 2k + 1 to 1?

7. ## Re: Number 1 Is Odd

2(0) + 1 = 1
2(1) + 1 = 3
2(2) + 1 = 5
2(3) + 1 = 7
.....
even + 1 = odd

8. ## Re: Number 1 Is Odd Originally Posted by DenisB 2(0) + 1 = 1
2(1) + 1 = 3
2(2) + 1 = 5
2(3) + 1 = 7
.....
even + 1 = odd
Good.