Find the remainder when 5

^{40}is divided with 11?
(1)1 (2) 0 (3)
2 (4) 10 (5) none of these

Solution
follows here:__Solution:__
Here we try
to convert the 5 powers in to multiple of 11. Then we apply binomial expansion,
to make the expression much simpler.

5

^{2}= 25 = 24+1, here 24 is not a multiple of 11
5

^{3}= 125 = 124+1, here 124 is not a multiple of 11
5

^{4}= 3125 = 3124+1, here 3124 is a multiple of 11 => 3124 = 11*284
5

^{40}= (5^{4})^{10}= (1+3124)^{10}= {1+(11*284)}^{10}
= 10C0 + 10C1 (11*284)

^{1}+ 10C2 (11*284)^{2}+…+10C10 (11*284)^{10}
= 1 + (multiple
of 11)

If this is divided with 11, the remainder will be 1

Answer (1)
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