Question 1210616
EDIT: My calculations are wrong. See The much clearer and direct solution by iklein.

For every 4 stamps donated by Emir, there were 7 stamps donated by Cillian.
For every 3 stamps donated by Cillian, there were 5 stamps donated by Lasha.
That means that for every {{{7*3=21}}} stamps given away by Cillian, there were
{{{4*3=12}}} stamps given away by Emir, and
This is wrong --> {{{5*3=15}}} stamps given away by Lasha.
It should have been {{{5*7=35}}} .
 
Let us make
the number of stamps donated by Emir {{{"="}}}{{{12x}}} ,
the number of stamps given by Cillian {{{"="}}}{{{21x}}} , and
the number of stamps given by Lasha {{{"="}}}{{{15x}}} ,
 
EDIT:Here is where the other solutions are better (besides having no mistakes). I insisted on calculating number of stamps kept based on {{{x}}} , but that unnecessarily complicates the calculations.

The number of stamps Emir gave away was 20 fewer than the number of stamps he kept for himself means Emir kept {{{12x+20}}} stamps. See the much clearer and direct solution by Iklein.

Cillian kept 3/2 as many stamps as Emir, Cillian and Lasha kept means
Cillian kept {{{(3/2)(12x+20)=18x+30}}} stamps.
 
Lasha kept 4/3 as many stamps as Cillian kept means
Lasha kept {{{(4/3)(18x+30)=24x+40}}} stamps.

After their donations, the total number of stamps Emir, Cillian and Lasha had  left in their collections was
{{{(12x+20)+(18x+30)+(24x+40)=738}}}<-->{{{(12+18+24)x+20+30+40=738}}}-->{{{54x+90=738}}}-->{{{54x=738-90}}}-->{{{54x=648}}}-->{{{x=648/54}}}-->{{{x=12}}}
 
Lasha gave away{{{15*12=highlight(180)}}} stamps.
 
In table form that looks like:
{{{matrix(4,4," ",Emir,Cillian, Lasha,
initial,"?","?","?",
donated,12x,21x, 15x,
kept,12x+20,(3/2)(12x+20),(4/3)(3/2)(12x+20))}}} --> {{{matrix(4,4," ",Emir,Cillian, Lasha,
initial,"?","?","?",
donated,12x,21x,15x,
kept,12x+20,18x+30,24x+40)
)}}} 
After solving {{{(12x+20)+(12x+20)+(24x+40)=738}}}->->-> {{{x=12}}} , we can substitute that value to get
{{{matrix(4,4," ",Emir,Cillian, Lasha,
initial,"?","?","?",
donated,12x,21x,15*12,
kept,164,246,328)
)}}} 
We can calculate the number of stamps each one originally had, and the number each donated,
but we are only asked for the number Lasha donated: {{{15*12=180}}}.