Actually here is a basic question, but i have a little problem about it.
In binary system, for any number such as 1011001, can we say directly "it is end with 1, so it is an odd number"?, or firstly should we convert it to decimal form, then look for is it odd or even number?
You can immediately conclude that a binary ending in 1 is odd, in the same way that a decimal number ending in an odd number is odd. A binary number of $n+1$ bits is in the form $$ a_n 2^n+...+a_3 2^3+a_2 2^2+a_1 2^1+a_0 2^0 $$ The last term is the "ones" place and all of the terms to the left of that are multiples of two (in the form $2^k$ for integer $k>1$). Therefor only $a_0$ in the right most term determines oddness. Of course by the same argument if it ends in $0$ then it is even.
Yes, it is the only bit where oddness can be added to a number. The other bits add even parts.
A number is odd if and only if its binary representation ends in a $1$.
This is equivalent to the fact that if a decimal number ends in $1,3,5,7$ or $9$, then it is odd.
