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Binomial Theorem question

2009 · Shift 0 · Q42
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Binomial Theorem question

2009 · Shift 0 · Q42

JEE MainMathematicsBinomial TheoremMCQ+4 / −1
The remainder left out when 82n−(62)2n+1{8^{2n}} - {\left( {62} \right)^{2n + 1}}82n−(62)2n+1 is divided by 9 is :
  1. A
    2
  2. B
    7
  3. C
    8
  4. D
    0
View written solutionFree

Correct answer: A

  1. We need the remainder when 82n−622n+18^{2n} - 62^{2n+1}82n−622n+1 is divided by 999.

So we work modulo 999.

  1. Reduce each base modulo 999: 8≡−1(mod9)8 \equiv -1 \pmod 98≡−1(mod9) and 62≡62−54=8≡−1(mod9).62 \equiv 62-54=8 \equiv -1 \pmod 9.62≡62−54=8≡−1(mod9).

  2. Now compute each term modulo 999:

  • First term: 82n≡(−1)2n=1(mod9)8^{2n} \equiv (-1)^{2n} = 1 \pmod 982n≡(−1)2n=1(mod9)

  • Second term: 622n+1≡(−1)2n+1=−1≡8(mod9)62^{2n+1} \equiv (-1)^{2n+1} = -1 \equiv 8 \pmod 9622n+1≡(−1)2n+1=−1≡8(mod9)

  1. Therefore, 82n−622n+1≡1−(−1)=2(mod9).8^{2n} - 62^{2n+1} \equiv 1 - (-1) = 2 \pmod 9.82n−622n+1≡1−(−1)=2(mod9). So the remainder is 2.2.2.

  2. Check options:

  • A: 222 ✅
  • B: 777 ❌
  • C: 888 ❌
  • D: 000 ❌

Hence the correct option is A.

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