Question

what is the remainder when 169(144^25) is divided by 13^4​

Answer 1

Answer:

Find the remainder when 169×14425 is divided by 134

Meanwhile I reduced it to

132×14425134=14425132=1250132

and Euler function of 132 is coming to be 156

I look for a short and simple way.

Step-by-step explanation:

The problem boils down to computing 14425(mod132), or 2525(mod132), or 2526(mod132), or 2513(mod132). By the binomial theorem:

2513=(26−1)13=∑k=013(13k)(−1)13−k(26)k

but for every k≥2 we have 26k≡0(mod132) and (131)(26)1≡0(mod132) too, so

2513≡−1(mod132),2526≡1(mod132),

14425≡−2525≡27(mod132)

and:

132⋅14425≡4563(mod134).