N*N mod 4 when N is odd

Number Theory Level 2

Given that $N$ is odd calculate $N\times N \bmod 4$ .

The answer is 1.

2 solutions

Chew-Seong Cheong
Nov 17, 2018

We can always express an odd integer as $N=2n+1$ , where $n$ is an integer. Then, we have:

$\begin{aligned} N^2 & \equiv 4n^2+4n+1 \equiv 0+0+1 \equiv \boxed 1\text{ (mod 4)}\end{aligned}$

Shouldn't you expand it as $4n^2 + 4n + 1$ ? In this way $4n^2 + 4n \equiv 0 \mod{4}$ .

Hans Gabriel Daduya - 2 years, 6 months ago

Thanks. I have changed it back. I was using the expansion earlier.

Chew-Seong Cheong - 2 years, 6 months ago

Srinivasa Gopal
Nov 17, 2018

Let N = (2p + 1) as N is odd Calculating N N as 4p p + 8p + 1 , we can see that the first two terms are fully divisible by 4, the last term leaves a reminder of 1 when divided by 4.

N*N mod 4 when N is odd

The answer is 1.

2 solutions

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