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)