remainder

note

32¹ leaves 4
32² leaves 2
32³ leaves 1
32⁴ leaves 4
32⁵ leaves 2
.
.
.
32³ⁿ⁺¹ will leave remainder 4

now considering division by 3
32¹ leaves 2
32² leaves 1
32³ leaves 2
.
.
32³² will leave 1 i.e. will be of the form 3n+1

so answer = 4

ya

thanx...was jsut verifying the answer..

ok [1]

Alternately we can also apply congruencies.....
32^{\phi(7)}\equiv 1(mod7) From which we have 32^{6}\equiv 1(mod7)
For the remaining part we have 32\equiv 2(mod6)
2^8=256\equiv 4(mod6)\implies 2^{32}\equiv 32^{32}\equiv 4(mod6)
Thus we just effectively need to figure out the rem when 32^4 is divided by 7, which is the rem when 256 is divided by 7, which is 4.