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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Basics of Mathematics

NDA 2024 · Mathematics · Basics of Mathematics

What is the remainder when 2^ 120 is divided by 7 ?

What is the remainder when 21202^{120} is divided by 7?7 ?

Options

  1. A1
  2. B3
  3. C5
  4. D6

Correct answer

A. 1

Step-by-step solution

2120=(23)40=840=(1+7)40=1+40C17+40C272+…+40C10770=1+7[40C1+40C27+…+40C40739]∴ Remainder =1.\begin{aligned} & 2^{120}=\left(2^3\right)^{40}=8^{40}=(1+7)^{40} \\ & =1+{ }^{40} \mathrm{C}_1 7+{ }^{40} \mathrm{C}_2 7^2+\ldots+{ }^{40} \mathrm{C}_{10} 7^{70} \\ & =1+7\left[^{40} \mathrm{C}_1+{ }^{40} \mathrm{C}_2 7+\ldots+{ }^{40} \mathrm{C}_{40} 7^{39}\right] \\ & \therefore \text{ Remainder }=1.\end{aligned}
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