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JEE MainMathematicsBinomial Theorem

The remainder when 13^ 14¹⁵ is divided by 11 is :

Options

  1. A1
  2. B2
  3. C9
  4. D5

Correct answer

D. 5

Step-by-step solution

We need to find the remainder when 13^ 14¹⁵ is divided by 11 . First, reduce the base modulo 11 : 13 2 11 So, we need to find 2^ 14¹⁵ 11 . By Fermat's Little Theorem, a^ p-1 1 p for a prime p . Here p = 11 , so: 2¹⁰ 1 11 This means we need to find the remainder of the exponent 14¹⁵ when divided by 10 . Let x = 14¹⁵ 10 . Since 14 4 10 , we have x 4¹⁵ 10 . Let us observe the powers of 4 modulo 10 : 4^1 4 10 4^2 = 16 6 10 4^3 = 64 4 10 4^4 = 256 6 10 The sequence of remainders is 4, 6, 4, 6, For any odd power, 4^ odd

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