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