Question
Find remainder when 7¹⁰⁰ is divided by
100.Solution
ATQ,
By Euler’s theorem, ϕ(100)=40 ⇒ 7⁴⁰ ≡ 1 mod 100. 100=40×2+20 ⇒ 7¹⁰⁰ ≡ (7⁴⁰)²×7²⁰ ≡ 1×7²⁰. 7⁴=2401≡1 mod100? 2401−2400=1 ⇒ 7⁴≡1. So 7²⁰=(7⁴)⁵≡1 ⇒ remainder 1.
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