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CAT 2022 Slot 2 · Quant · Number System · hard

What is the remainder when 7¹⁰⁰ is divided by 25?

Single correct option

What is the remainder when 7¹⁰⁰ is divided by 25?
  1. A. 1
  2. B. 6
  3. C. 7
  4. D. 24
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Answer: (A) 1

By Euler's theorem, since gcd(7,25)=1, 7^φ(25) ≡ 1 (mod 25), where φ(25) = 25×(1−1/5) = 20. So 7²⁰ ≡ 1 (mod 25). Since 100 = 20×5, 7¹⁰⁰ = (7²⁰)⁵ ≡ 1⁵ = 1 (mod 25). The remainder is 1.
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Original question written in the CAT 2022 Slot 2 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake