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16 Sep 2014, 00:40
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16 Sep 2014, 00:40
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Official Solution:
What is the last digit of \(3^{3^3}\)?
A. 1
B. 3
C. 6
D. 7
E. 9
THEORY
When exponentiation is represented by stacked symbols, the rule is to work from the top down. For example, \(a^{m^n}=a^{(m^n)}\) and not \((a^m)^n\), which instead equals \(a^{mn}\). Therefore:
\((a^m)^n=a^{mn}\).
\(a^{m^n}=a^{(m^n)}\).
BACK TO THE QUESTION
Using the above rule, \(3^{3^3}=3^{(3^3)}=3^{27}\).
Next, the units digit of 3 in positive integer power follows a repeating pattern of 4 digits: {3, 9, 7, 1}. Hence, we can determine the units digit of \(3^{27}\) by finding the 27th digit in this sequence, which is 7. (This is because 27 divided by the cyclicity number of 4 gives a remainder of 3, and thus, the units digit of \(3^{27}\) will correspond to the third number in the pattern, which is the same as the units digit of \(3^3\).)
Answer: D
What is the last digit of \(3^{3^3}\)?
A. 1
B. 3
C. 6
D. 7
E. 9
THEORY
When exponentiation is represented by stacked symbols, the rule is to work from the top down. For example, \(a^{m^n}=a^{(m^n)}\) and not \((a^m)^n\), which instead equals \(a^{mn}\). Therefore:
\((a^m)^n=a^{mn}\).
\(a^{m^n}=a^{(m^n)}\).
BACK TO THE QUESTION
Using the above rule, \(3^{3^3}=3^{(3^3)}=3^{27}\).
Next, the units digit of 3 in positive integer power follows a repeating pattern of 4 digits: {3, 9, 7, 1}. Hence, we can determine the units digit of \(3^{27}\) by finding the 27th digit in this sequence, which is 7. (This is because 27 divided by the cyclicity number of 4 gives a remainder of 3, and thus, the units digit of \(3^{27}\) will correspond to the third number in the pattern, which is the same as the units digit of \(3^3\).)
Answer: D
10 Apr 2016, 04:26
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