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I came with this while playing with numbers.

after i divide $6$ into two parts say $3$ x $2$ i continue like that

  1. i divide first number with some random number $3:2 = 1,5$
  2. then multiply second one with the same random number $2$ x $ 2 = 4$

3.Now i multiply both result numbers $1,5$ x $4$ and the result is the number which i begin with $ = 6$

Random numer needs to be the same in both steps (1&2).
What is the name operation im performing and where are common applications of this operation?

Looks like some kind of factoring to me hence the tag.

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    This is obvious, since for every non-zero $x$ that you choose: $a\cdot b=(a/x)\cdot(b\cdot x)$. The names of the operations that you are performing are: multiplication, division and random-number generating. – 2017-01-11

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What you're doing is essentially saying that, given $a\times b$, you can rewrite it as: $$2a\times\frac{b}{2}$$ The $2$ here is arbitrary, for any number $r$ you can rewrite it as: $$ra\times\frac{b}{r}$$ This isn't an "operation", as it doesn't change the value of the thing you're working with (so it "acts as the identity" on all numbers). It can still be useful for mental arithmetic though. As an example: $$40\times 57 = 20\times (2\times 57) = 20\times 114 = 10\times 228 = 2280$$ Of course, this "mental trick" isn't necessairly easier than writing: $$40\times 57 = 40\times (50+7) = 2000+280 = 2280$$

This "dividing and multiplying by the same number" can be used fairly frequently in basic algebraic computations. Something I can easily see doing is writing that: $$2x+3+16x^2 = 2(x+\frac{3}{2}+8x^2)$$ Here, I wrote $3 = (2\times 1)\frac{3}{2}$ to factor a $2$ out of $3$. Of course, I don't really do this explicitly ever, but it happens "in the background".

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    are there any similiar "operations" to this ? – 2017-01-11
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    I'd be hesitant to call this an operation. Generally, an operation takes a certain number of objects, and returns a single object. An example of this is "$+$" takes two numbers, and returns $a+b$, a third number. Operations that work with a single input exist, so that isn't the issue here. The issue is that we're not changing anything really, just rewriting. So, this operation is really just the identity operation on all inputs. – 2017-01-12