Let n be any positive integer. Show that $2n - 1 \equiv -1\pmod n$

Question

I'm a very slow learner in math. Would someone be kind enough to explain in detail how to solve the above question?

Thanks in advance.

Answer 1

You have addition and multiplication modulo $n$. To calculate $2n-1\mod n$, you may calculate $2n\mod n$ and $-1\mod n$ separately and then add the results (and take the result $\mod n$ again, if necessary).

Thus, $2n-1\mod n = ((2n\mod n)+(-1\mod n))\mod n= (0-1)\mod n \equiv -1$.

Does this answer your question?

Answer 2

Consider basic definition of congruence:

If $A= 2\times n +r$

Then:

$A ≡ r\mod n$

Where r is remainder which in you question is $r=-1$

So $A=2\times n -1 ≡ -1\mod n$