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
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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- Max@Math Revolution
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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
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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