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Que: If n is positive integers, what is the unit digit of \(\left(3^{4n+1}\right)\left(9^{19}\right)\)?

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by Max@Math Revolution » Mon May 10, 2021 12:34 am

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Que: If n is positive integers, what is the unit digit of \(\left(3^{4n+1}\right)\left(9^{19}\right)\)?

A. 3
B. 7
C. 6
D. 2
E. 8
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Re: Que: If n is positive integers, what is the unit digit of \(\left(3^{4n+1}\right)\left(9^{19}\right)\)?

by Max@Math Revolution » Mon May 10, 2021 9:20 pm
Solution: Powers that repeat every 4th power:

Ex) Units digit: The digit 3 repeats after every fourth power

=> \(3^1\)= ~3, ~\(3^2\)= ~9, ~\(3^3\) = ~7, ~\(3^4\)= ~1, ~\(3^5\)= ~3, ...

=> Pattern: 3, 9, 7, 1, 3, 9, 7, 1 ….

Ex) Units digit: The digit 9 repeats after every second power

=> ~\(9^1\) = ~9, ~\(9^2\) = ~1, ~\(9^3\) = ~9, ... => Pattern: 9, 1, 9, 1…

We have to find the units digit of \(\left(3^{4n+1}\right)\left(9^{19}\right)\)if n is positive integers

=> \(\left(3^{4n+1}\right)\left(9^{19}\right)\)

=> \(\left(3^4\right)^n\cdot3^1\cdot\left(~9\right)\)

=> \(\left(81\right)^n\cdot3^1\cdot\left(~9\right)=\left(~1\right)\cdot3^1\cdot\left(~9\right)=\left(~7\right)\)

Therefore, B is the correct answer.

Answer B
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