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# M09-32

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Math Expert
Joined: 02 Sep 2009
Posts: 56275

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16 Sep 2014, 00:40
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3
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Difficulty:

45% (medium)

Question Stats:

50% (00:22) correct 50% (00:23) wrong based on 136 sessions

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What is the last digit of $$3^{3^3}$$?

A. 1
B. 3
C. 6
D. 7
E. 9

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Math Expert
Joined: 02 Sep 2009
Posts: 56275

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16 Sep 2014, 00:40
1
1
Official Solution:

What is the last digit of $$3^{3^3}$$?

A. 1
B. 3
C. 6
D. 7
E. 9

THEORY

If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus:

$$a^{m^n}=a^{(m^n)}$$ and not $$(a^m)^n$$, which on the other hand equals to $$a^{mn}$$.

So:

$$(a^m)^n=a^{mn}$$;

$$a^{m^n}=a^{(m^n)}$$ and not $$(a^m)^n$$.

BACK TO THE QUESTION

According to the above: $$3^{3^3}=3^{(3^3)}=3^{27}$$

Next, the units digit of 3 in positive integer power repeats in pattern of 4: {3, 9, 7, 1}. So, the units digit of $$3^{27}$$ (27=4*6+3) is the same as the units digit of $$3^3$$, which is 7.

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Intern
Joined: 04 Mar 2015
Posts: 6

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10 Apr 2016, 00:17
(27=4*6+3) so you add 4 + 3 = 7 (Please more details)
Math Expert
Joined: 02 Sep 2009
Posts: 56275

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10 Apr 2016, 04:26
Senior Manager
Joined: 15 Feb 2018
Posts: 258

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06 Sep 2018, 01:16
I think this is a high-quality question and I agree with explanation. You have it as a Fractions/Ratios/Decimals question when it should be Exponents and Roots?
Intern
Joined: 30 Aug 2017
Posts: 10
GMAT 1: 580 Q45 V26
GMAT 2: 700 Q48 V38
GMAT 3: 710 Q49 V39

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17 Nov 2018, 08:03
I think this is a high-quality question and I agree with explanation.
Intern
Joined: 06 Aug 2012
Posts: 1

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13 Jan 2019, 01:40
I am not clear with the explanation as I know
3^3 * 3^3 * 3^3 = (3^3)^3 = 3^9

The question has (3^3)^3 what is equal to 3^9 and the unit digit of 3^9 is 3

Please give a details explanation with example. Thanks
Math Expert
Joined: 02 Sep 2009
Posts: 56275

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13 Jan 2019, 03:17
smoni wrote:
I am not clear with the explanation as I know
3^3 * 3^3 * 3^3 = (3^3)^3 = 3^9

The question has (3^3)^3 what is equal to 3^9 and the unit digit of 3^9 is 3

Please give a details explanation with example. Thanks

I think you did not read the discussion above carefully enough.

The question does not ask about the units digit of $$(3^3)^3$$, it asks about the units digit of $$3^{3^3}$$. The solution clearly explains the difference between these two expression. Please re-read.
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Re: M09-32   [#permalink] 13 Jan 2019, 03:17
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# M09-32

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