Question

find the remainder of

\(\large\tt \color{black}{50^{51^{52}}}\) divided by 11

Answer 1

hmm i wish there is new NT problems

Answer 2

I want to avoid using NT stuff as much as possible here because this question is for entrance tests :

First, notice that 51^52 leaves a remainder of 1 when divided by 10

Answer 3

yes

Answer 4

you can figure that out by using binomial theorem :

51^52 = (50+1)^52 = 10M + 1

Answer 5

yes

Answer 6

50^51^52 = 50^(10M+1)

agree ?

Answer 7

yes

Answer 8

if you heard of Fermat little theorem before :
50^10 leaves a remainder of 1 when `divided by 11`

50^51^52 = 50^(10M+1)
= 50*50^(10M)
= 50 * 1
= 50
= 6

so the remainder is 6

Answer 9

we need to apply binomial theorem again if you don't want to use Fermat little theorem

Answer 10

fermat rule says 50^(11n) mod 11=1 ?

Answer 11

a^kn mod k=1

Answer 12

it should be a^(k-1) mod k congruent to 1

Answer 13

ok thanks very much

Answer 14

so u got it ?

Answer 15

\(\large\tt \color{black}{a^{p-1}\bmod{p}\equiv 1\bmod{p}}\)

Answer 16

right?

Answer 17

i think i got it