Tuesday 20 December 2011

Numbers -25

Find the remainder when 540 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.
52 = 25 = 24+1, here 24 is not a multiple of 11
53 = 125 = 124+1, here 124 is not a multiple of 11
54 = 3125 = 3124+1, here 3124 is a multiple of 11 => 3124 = 11*284
540 = (54)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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