1. Modular Arithmetic (Basic)

prove that a( a + 1 )(2a + 1) is divisible by 6 for every integer a.

Now my book says

"By taking least absolute residues mod (6), we see that a = 0, -1,1 -2,2 or 3"

What the hell does "least absolute residue" mean? Can someone explain this very carefully? (Yes I'm self learning so please don't leave out any important details)

And how do they get a = 0,-1,1,-2,2 or 3???????

Thanks very much!!!

2. If this is divisible by $6$, then we have :

$a(a + 1)(2a + 1) \equiv 0 \pmod{6}$

Note that from the definition of the modulus operation, $a \in [0..5]$ (try with $a = 6$, you will get the same result than with $a = 0$). That comes from the fact that $ab \equiv (a \pmod{m})(b \pmod{m}) \pmod{m}$. Thus you can check all values of $a \in [0..5]$ :

$a = 0$, $0(0 + 1)(2 \times 0 + 1) \equiv 0 \pmod{6}$

$a = 1$, $1(1 + 1)(2 \times 1 + 1) \equiv 1 \times 2 \times 3 \equiv 6 \equiv 0 \pmod{6}$

Etc ... and you will find out that for $a \in [0..5]$, it works. Thus it works for $a + 6k$, with $k \in \mathbb{Z}$, which matches the set of all integer $a$.

Is it clear enough, or didn't you understand anything ?

3. Originally Posted by Bacterius
If this is divisible by $6$, then we have :

$a(a + 1)(2a + 1) \equiv 0 \pmod{6}$

Note that from the definition of the modulus operation, $a \in [0..5]$ (try with $a = 6$, you will get the same result than with $a = 0$). That comes from the fact that $ab \equiv (a \pmod{m})(b \pmod{m}) \pmod{m}$. Thus you can check all values of $a \in [0..5]$ :

$a = 0$, $0(0 + 1)(2 \times 0 + 1) \equiv 0 \pmod{6}$

$a = 1$, $1(1 + 1)(2 \times 1 + 1) \equiv 1 \times 2 \times 3 \equiv 6 \equiv 0 \pmod{6}$

Etc ... and you will find out that for $a \in [0..5]$, it works. Thus it works for $a + 6k$, with $k \in \mathbb{Z}$, which matches the set of all integer $a$.

Is it clear enough, or didn't you understand anything ?
Thank you, that was so helpful!!

4. Alternatively:

$1^2+2^2+\ldots + a^2 = \frac{a(a+1)(2a+1)}{6}$

5. Alternatively (alternatively):

Let $f(n)=n(n+1)\left(2n+1\right)$

Then $f(n+1)-f(n)=6\left(n+1\right)^2$ and the it follows by induction.