Attempt this Number System Practice Question

Question of The Day13-12-2021

Find the remainder when 4²⁰ + 16²⁰ + 64²⁰ + 256²⁰ is divided by 4¹⁹ – 1

Answer

Correct Answer : c ) 340

Explanation :

According to the question, we have to find the remainder in the following division:

\({40^{20}+16^{20}+64^{20}+256^{20}}\over {4^{19}-1}\)

\(=>{{4^{20}+{(4^{20})}^2+{(4^{20})^3}+{(4^{20})4}}\over {4^{19}-1}}\)

4²⁰ = 4¹⁹ * 4 + 4 – 4

4²⁰ = 4(4¹⁹ – 1) + 4

\(=>{{4 {(4^{19}-1)}+4}\over {4^{19}-1}}\)

⇒remainder = 4

\(=>{{4^{20}}\over {4^{19}-1}}+{{(4^{20})^2}\over {4^{19}-1}}+{{(4^{20})^3}\over {4^{19}-1}}+{{(4^{20})^4}\over {4^{19}-1}}\)

⇒Remainder = (4) + (4)² + (4)³ + (4)⁴\

⇒ 4 + 16 + 64 + 256

⇒340

Hence, (c) is the correct answer.

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