# Thread: Divisibility 11

1. ## Divisibility 11

Show that: $\displaystyle a, k\in \mathbf{Z},$

$\displaystyle a=2k \Leftrightarrow a-2\| \frac{a}{2}\| = 0$

.............................................

Symbol...
Example:

$\displaystyle \| 5,3\|=5$

$\displaystyle \| -2,2\|=-3$

2. Originally Posted by Sea
Show that: $\displaystyle a, k\in \mathbf{Z},$

$\displaystyle a=2k \Leftrightarrow a-2\| \frac{a}{2}\| = 0$

.............................................

Symbol...
Example:

$\displaystyle \| 5,3\|=5$

$\displaystyle \| -2,2\|=-3$
In more familiar notation:

Show that: $\displaystyle a, k\in \mathbb{Z},$

$\displaystyle a=2k \Leftrightarrow a-2 \left\lfloor \frac{a}{2}\right\rfloor = 0$

where $\displaystyle \lfloor x \rfloor$ is the floor function, the greatest integer less than $\displaystyle x$.

CB

3. Originally Posted by CaptainBlack
In more familiar notation:

Show that: $\displaystyle a, k\in \mathbb{Z},$

$\displaystyle a=2k \Leftrightarrow a-2 \left\lfloor \frac{a}{2}\right\rfloor = 0$

where $\displaystyle \lfloor x \rfloor$ is the floor function, the greatest integer less than $\displaystyle x$.

CB
Suppose $\displaystyle a=2k$ for some $\displaystyle k \in \mathbb{Z}$ then:

$\displaystyle a-2\left\lfloor \frac{a}{2}\right\rfloor=a-2 \lfloor k \rfloor=a-2k=0$

Which proves

$\displaystyle a=2k \Rightarrow a-2 \left\lfloor \frac{a}{2}\right\rfloor = 0$

To complete the proof assume that: $\displaystyle a-2 \left\lfloor \frac{a}{2}\right\rfloor = 0$ and show that this implies that there exists a $\displaystyle k \in \mathbb{Z}$ such that $\displaystyle a=2k$

CB

4. I didn't know this words means in english...I'm sorry for this and I thank you so much... for answering my question after all...

Thanks a million......