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The remainder when 4⁰ + 4¹ + 4² + 4³ + ........ + 4⁴⁰ is divided by 17 is:

10
216
34
4None of these

Answer:- 1

Explanation:-

Solution:

Let S be the sum of the expression, Then
S = 4⁰ + 4¹ + 4² + 4³ + ........ + 4⁴⁰
S = (1 + 4 + 16 + 64) + 4⁴ (1 = 4 + 16 + 64) + ...... + 4³⁶ + 4⁴⁰
Since, (1 + 4 + 16 + 64) = 85, is divisible by 17. Hence, except 4⁴⁰ remaining expression is divisible by 17.
Thus,

\frac{4^{40}}{17} \to \frac{{(4^{4})}^{10}}{17} \to \frac{1^{10}}{17}

$\frac{{{4^{40}}}}{{17}} \to \frac{{{{\left( {{4^4}} \right)}^{10}}}}{{17}} \to \frac{{{1^{10}}}}{{17}}$
Hence, remainder = 1

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Hence, remainder = 1

", "dateCreated": "7/24/2019 10:09:12 AM", "author": { "@type": "Person", "name": "Nitin Sir" } }, "suggestedAnswer": { "@type": "Answer", "text": "

Solution:

Let S be the sum of the expression, Then
S = 4⁰ + 4¹ + 4² + 4³ + ........ + 4⁴⁰
S = (1 + 4 + 16 + 64) + 4⁴ (1 = 4 + 16 + 64) + ...... + 4³⁶ + 4⁴⁰
Since, (1 + 4 + 16 + 64) = 85, is divisible by 17. Hence, except 4⁴⁰ remaining expression is divisible by 17.
Thus,

\frac{4^{40}}{17} \to \frac{{(4^{4})}^{10}}{17} \to \frac{1^{10}}{17}

$\frac{{{4^{40}}}}{{17}} \to \frac{{{{\left( {{4^4}} \right)}^{10}}}}{{17}} \to \frac{{{1^{10}}}}{{17}}$
Hence, remainder = 1

", "dateCreated": "7/24/2019 10:09:12 AM" } }

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