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