Question

how to calculate 6^7^8^9 mod 13, whithout
calculator

Answer 1

i would do them one at at time, but i bet there is a snappier method

Answer 2

6^7 = (2^7)(3^7) = 6(64)(27^2)

6 = 6mod13
64=-1mod13
27=1mod13 =>27^2 = (1^2)mod13 = 1mod13

=>(6*64*27^2) = -6mod13
=>(6^7) = -6mod13

(6^7^8) = ((-6)^8)mod13
(-6)^8 = 6^8 = 2^8 * 3^8 = 36.(64).(27^2)

following same procedure,we will get

((-6)^8)mod13 = 3mod13

can you do the next step ?

Answer 3

there are lots of ways to compute \(6^7\) mod 13

Answer 4

(6^7^8^9) =3^9 (mod 13)

27^3 = 1^3 (mod 13) = 1 (mod 13)

...so the solution is 1?

Answer 5

i think 1 is right, yes

Answer 6

So the procedure is: you take 6^7 and calculate its modulus, then you you take modulus of 6^7 to the power of 8 calculate its modulus, and repeat again for the 9?