Question 1207719
.
Let m and n be non-negative integers. If m = 6n + 2, then what integer between 0 and m 
is the inverse of 2 modulo m? Answer in terms of n.
~~~~~~~~~~~~~~~~~~~~~~


<pre>
Notice that m = 6n+2 is an even integer number, for any integer number n.


Let's assume that an integer x between 0 and m is the inverse of 2 modulo m.


It means that 2*x = 1 mod m, which is the same as to say that

    2x - 1 is a multiple of m :   2x - 1 = k*m.


But 2x is an even number, and k*m is an even number, since "m" is even.


Therefore, this equality  2x - 1 = km  is not possible with integer x and "k".


Hence, there is NO any integer between 0 and m which is inverse of 2 modulo m.


<U>ANSWER</U>.  There is NO any integer between 0 and m which is inverse of 2 modulo m.
</pre>

Solved.



---------------



What I proved in my post, &nbsp;in terms of abstract algebra is &nbsp;THIS &nbsp;general statement:



    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In the ring of integers modulo &nbsp;m,  &nbsp;&nbsp;Z/m,  &nbsp;&nbsp;where &nbsp;" m " &nbsp;is an even number,

    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the class &nbsp;{2 mod m} &nbsp;is &nbsp;NOT &nbsp;an invertible element.



In opposite, in such a ring, &nbsp;the class  &nbsp;{2 mod m}  &nbsp;is a divisor of zero;
and it is well known fact of abstract algebra that in a ring a divisor of zero can not be invertible.