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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