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📐 Mathematics  ·  Binomial Theorem  ·  JEE

Using binomial theorem, find the remainder when 3¹⁰⁰ is divided by 4.

Answer: 1.

  • A 0
  • B 2
  • C 3
  • D 1

Correct answer: D. 1

Explanation: 3<sup>100</sup> = (4-1)<sup>100</sup> = 4<sup>100</sup> - 100 x 4<sup>99</sup> + ... + (-1)<sup>100</sup> = 4<sup>100</sup> - ... + 1. All terms except the last contain factor 4. So 3<sup>100</sup> mod 4 = 1.

Pascal's Triangle: Binomial Coefficients11112113311464115101051n=0n=1n=2n=3n=4n=5Each entry = sum of the two entries diagonally above it (Pascal's identity); row n gives nC0...nCn

Row n of Pascal's triangle gives the coefficients nC0, nC1, ..., nCn for the expansion of (a+b)ⁿ; each number is the sum of the two numbers diagonally above it, a direct visual proof of the identity nCr = (n-1)C(r-1) + (n-1)Cr.

Concept context

Expansion of (a+b) n , general term, middle term, binomial coefficients, and greatest term

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