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# What is the last digit of 2^3^4?

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What is the last digit of 2^3^4? [#permalink]  07 Sep 2012, 05:17
00:00

Difficulty:

15% (low)

Question Stats:

66% (01:34) correct 34% (00:19) wrong based on 75 sessions
What is the last digit of $$2^{3^4}$$ ?

(A) 0
(B) 2
(C) 6
(D) 4
(E) 8
[Reveal] Spoiler: OA

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Last edited by Bunuel on 07 Sep 2012, 05:38, edited 1 time in total.
Renamed the topic.
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Re: What is the last digit of 2^3^4? [#permalink]  07 Sep 2012, 05:47
Expert's post
What is the last digit of $$2^{3^4}$$ ?

(A) 0
(B) 2
(C) 6
(D) 4
(E) 8

If exponentiation is indicated by stacked symbols, the rule is to work from the top down: $$a^m^n=a^{(m^n)}$$ and not $$(a^m)^n$$. For more check Number Theory chapter of Math Book: math-number-theory-88376.html

Hence, $$2^{3^4}=2^{(3^4)}=2^{81}$$.

Now, the units digit of 2 in positive integer power repeats the following pattern {2, 4, 8, 6}:
2^1=2;
2^2=4;
2^3=8;
2^4=16;
2^5=32 (2 again),

Therefore the units digit of $$2^{81}$$ will be the same as the units digit of 2^1 (81=4*20+1), so 2.

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Re: What is the last digit of 2^3^4? [#permalink]  07 Sep 2012, 09:00
What is the last digit of $$2^{3^4}$$ ?

(A) 0
(B) 2
(C) 6
(D) 4
(E) 8

This kind of "what is last digit" problems can be easily solved using rule of cyclicity.

We need to remember that cyclicity of 2, 3, 7, 8 is 4. This means original number repeats after 4th power, hence $$2^5$$ has first digit 2.
Cyclicity of 4, 9 is 2.
1, 5, 6 will remain same always.

With this basic knowledge we can proceed to answer this question.
$$2^{3^4}$$
$$=2^{81}$$
$$->2^1$$

So, last digit is 2.

With this method, I am sure one can answer such question within 1 minute.
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Re: What is the last digit of 2^3^4?   [#permalink] 07 Sep 2012, 09:00
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