Question #199455

Find the remainder when 7^21+49^21+343^21+2401^21 is divided by 7^20+1


1
Expert's answer
2021-06-03T08:12:38-0400

1) 721+4921+34321+240121=7^{21} + 49^{21} + 343^{21} + 2401{21} =

721+(77)21+(777)21+(7777)21=7^{21} + (7\cdot7)^{21}+(7\cdot7\cdot7)^{21} + (7\cdot7\cdot7\cdot7)^{21} =

721+(721)2+(721)3+(721)4;7^{21} +{(7^{21})}^2+{(7^{21})}^3 + {(7^{21})}^4;

2) 721=7(720)=7(720+1)77^{21} = 7\cdot(7^{20})=7\cdot({7^{20} + 1})-7

3) Rem[721/(720+1)]=7Rem[7^{21} / (7^{20} + 1)] = -7

We have represented 721 as 7⋅(720+1)−7 (look step 2)

7⋅(720+1)−7 mod (720+1) = -7 because


7(721+1)7721+1=7(721+1)721+1+7721+1=7+7721+1\frac{7\cdot(7^{21}+1) -7}{7^{21}+1} = \frac{7\cdot(7^{21}+1)}{7^{21}+1}+\frac{-7}{7^{21}+1} = 7+\frac{-7}{7^{21}+1}


where is -7 is our reminder.

For steps 4, 5, 6 We have to raise our reminder to a power.

4) Rem[(721)2/(720+1)]=(7)2Rem[{(7^{21})}^2/(7^{20} + 1)] = (-7)^{2}

5) Rem[(721)3/(720+1)]=(7)3Rem[{(7^{21})}^3/(7^{20} + 1)] = (-7)^3

6) Rem[(721)4/(720+1)]=(7)4Rem[{(7^{21})}^4/(7^{20} + 1)] = (-7)^4

Answer:7+(7)2+(7)3+(7)4=2100Answer: -7+(-7)^2+(-7)^3+(-7)^4 = 2100



Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS