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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Binomial Theorem

NDA 2024 · Mathematics · Binomial Theorem

What is the remainder when 7^n-6 n is divided by 36 for n=100 ?

What is the remainder when 7n−6n7^n-6 n is divided by 36 for n=100n=100 ?

Options

  1. A00
  2. B11
  3. C22
  4. D66

Correct answer

B. 11

Step-by-step solution

Suppose p(n)=7n−6n=(6+1)n−6nUsingBinomialexpansion,weget\mathrm{p}(\mathrm{n})=7^{\mathrm{n}}-6 \mathrm{n}=(6+1)^{\mathrm{n}}-6 \mathrm{n} Using Binomial expansion, we get=1+6n+nc262+nc363+…..+6n−6n=1+62(nc2+nc2×6+….+6n−2)=1+36(100c2+100c2×6+….+698){∵n=100}=1+36 m where  m=100c2+100c2×6+…..+698\begin{aligned} & =1+6 n+{ }^n c_2 6^2+{ }^n c_3 6^3+\ldots . .+6^n-6 n \\ & =1+6^2\left({ }^n c_2+{ }^n c_2 \times 6+\ldots .+6^{n-2}\right) \\ & =1+36\left({ }^{100} c_2+{ }^{100} c_2 \times 6+\ldots .+6^{98}\right)\{\because n=100\} \\ & =1+36 \mathrm{~m} \text{\text{ where } } \mathrm{m}={ }^{100} c_2+{ }^{100} c_2 \times 6+\ldots . .+6^{98} \end{aligned}So,whenSo, when7^n{\mathrm{n}}-6 n\mathrm{n} is divided by 36 , the remainder will be 1.
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