MHB Transporting 500^32 Apples to 6 Cities: How Many Remain?

[Albert1]

$ If\,\,you\,\,equally\,\,transport \,\,32^{500} \,\,apples \, \,among \,\,6\,\, cities$

$how\,\,many\,\,apples\,\ will\,\, remain \,\,?$

 

[kaliprasad]

$ If\,\,you\,\,equally\,\,transport \,\,32^{500} \,\,apples \, \,among \,\,6\,\, cities$

$how\,\,many\,\,apples\,\ will\,\, remain \,\,?$

Spoiler

we need $32^{500} \pmod {6}$

so let us find $32^{500} \pmod {2}$ and $32^{500} \pmod {3}$
the 1st part is zero and 2nd part is

$32^{500} \pmod {3} = 2^{500} \pmod {3} =(2^{2})^{250} \pmod {3}$
$= (4)^{250} \pmod {3} = (1)^{250} \pmod {3} = 1$
we need to solve

$x \pmod {3} =1$ and $x \pmod {2} = 0$ giving 4
so ans is 4