Argue why the following statements hold n2n 1 4n 1 mod 3

Argue why the following statements hold. n(2n + 1) (4n + 1) = (mod 3) Squareroot 3 is not a rational number.

Solution

1.

We know 2=-1 mod 3,4=1 mod 3

So we can rewrite expression as

n(-n+1)(n+1)=-(n-1)n(n+1)

So it reduces to product of three consecutive integers. One of them must be a multiple of 3.

Hence, n(2n+1)(4n+1)=0 mod 3

2.

Let it be rational

So,

sqrt{3}=p/q where, p,q integers, gcd(p,q)=1

Squaring gives

3q^2=p^2

Hence, 3|p^2 and 3 is prime so ,3|p

Hence, p=3m for some integer m

So,

3q^2=3m^2

q^2=3m^2

Again we have :3|q^2 hence, 3|q

But, gcd(p,q)=1 but we have:3|p,3|q

Hence a contradiction

So, sqrt{3} is irrational.