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

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by Max@Math Revolution » Mon Mar 15, 2021 9:07 pm

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

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

by Max@Math Revolution » Thu Mar 18, 2021 9:19 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 4 repeats after every second power

=> ~\(4^1\) = ~4, ~\(4^2\) = ~6, ~\(4^3\) = ~4, ...

=> Pattern: 4, 6, 4, 6…

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

=> \(\left(3^{4n+1}\right)\cdot\left(4^{19}\right)\)
=> \(\left(3^4\right)^n\cdot3^1\cdot\left(~4\right)\)
=> \(\left(81\right)^n\cdot3^1\cdot\left(~4\right)\) =(~1)* 3^1 *(~4) =(~2)

Therefore, B is the correct answer.

Answer B
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