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M13-36

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Bunuel
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M13-36 [#permalink] New post   16 Sep 2014, 00:50
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If \(x\) is a positive integer, what is the hundreds digit of \(x^x\)?


(1) \(x\) is divisible by 10.

(2) \(x\) is even.
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Bunuel
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Re M13-36 [#permalink] New post   16 Sep 2014, 00:50
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Official Solution:


If \(x\) is a positive integer, what is the hundreds digit of \(x^x\)?

(1) \(x\) is divisible by 10.

This implies that \(x\) could be 10, 20, 30, and so on. Thus, \(x^x\) could be \(10^{10}\), \(20^{20}\), \(30^{30}\), etc. The hundreds digit of any of these numbers will be 0, as each of these numbers ends with at least ten zeros (because they have factors of both \(5^{10}\) and \(2^{10}\)). Sufficient.

(2) \(x\) is even.

For \(x=4\), \(4^4 = 256\), so the hundreds digit is 2. However, for \(x=10\), \(10^{10}=10,000,000,000\), so the hundreds digit is 0. Not sufficient.


Answer: A
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Re: M13-36 [#permalink] New post   14 Dec 2014, 17:06
since 0 is also divisible by 10, can we not have x = 0?
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Re: M13-36 [#permalink] New post   15 Dec 2014, 08:15
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kritiu wrote:
since 0 is also divisible by 10, can we not have x = 0?


We are given that x is a positive integer and 0 is neither positive nor negative integer.
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Re: M13-36 [#permalink] New post   17 Dec 2014, 14:48
Couldn't x = 10 (10 is divisible by 10)? If so, the hundreds digit would be 1 (10^10 = 100). Any insight would be helpful...
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Re: M13-36 [#permalink] New post   17 Dec 2014, 21:18
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mclark674162 wrote:
Couldn't x = 10 (10 is divisible by 10)? If so, the hundreds digit would be 1 (10^10 = 100). Any insight would be helpful...



X can be any positive multiple of 10 so yes if x=10 then the expression will be 10^10 and hundred digit will be 0 because 10^10= 10*10*10*10*10...10 times= 10000000000

Consider x=20,even then Hundreds digit will be 0
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Re: M13-36 [#permalink] New post   01 Aug 2018, 05:57
what if 0^0, how do we get the 100th digit?

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Re: M13-36 [#permalink] New post   01 Aug 2018, 06:11
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gsingh0711 wrote:
what if 0^0, how do we get the 100th digit?

Bunuel


We are given that x is a positive integer, so x cannot be 0. 0 is neither positive nor negative integer.

As for 0^0: 0^0, in some sources equals to 1, some mathematicians say it's undefined. But you won't need this for the GMAT because the case of 0^0 is not tested on the GMAT.
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Re: M13-36 [#permalink] New post   23 Oct 2023, 08:22
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I've revised the question and solution, incorporating additional details for improved clarity. I trust this makes it more comprehensible.
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Re: M13-36 [#permalink]
23 Oct 2023, 08:22
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