Why remainders of division give base expression? $214 = 3 · 71 + 1$
$71 = 3 · 23 + 2$
$ 23 = 3 · 7 + 2 $
$7 = 3 · 2 + 1 $
$2 = 3 · 0 + 2$
As a result, to obtain a base $3$ expansion of $214$, we take the remainders of divisions and we get that $(214)_{10} = (21221)_3$.
Question: why do the remainders give base $3$ expansion of $214$? And why order is from bottom to top?

By definition of the base $3$ numeration,
$$|abcde|_3=a\cdot3^4+b\cdot3^3+c\cdot3^2+d\cdot3+e.$$
Divide by $3$ to get the remainder $e$ and the quotient 
$$a\cdot3^3+b\cdot3^2+c\cdot3+d.$$
Divide by $3$ to get the remainder $d$ and the quotient 
$$a\cdot3^2+b\cdot3+c.$$
Divide by $3$ to get the remainder $c$ and the quotient 
$$a\cdot3+b.$$
Divide by $3$ to get the remainder $b$ and the quotient 
$$a.$$
Divide by $3$ to get the remainder $a$ and the quotient 
$$0.$$
