Find all $n\in\mathbb{Z}^+$ such that the sum of the digits of $5^n$ equals $2^n$   Starting with a table of values, I found that $n=3$ works. Beyond this, it's hard to imagine any other number working. As $n$ gets larger, it seems to me that $2^n$ grows far more rapidly than the sum of the digits of $5^n$ would ever be able to "catch up" to.  My reasoning: multiplying by each $2$ doubles $2^n$, but multiplying by each $5$ adds (I think) at most $1$ digit to $5^n$, which would add at most $9$ to the sum of the digits of $5^n$. Another way of looking at it, suppose $n=1000000$. Notice that $5^{1000000}&lt;10^{1000000}$, which is $1$ with $1000000$ zeros. Thus $5^{1000000}$ has at most $1000000$ digits, and the sum of its digits would be maximized if those digits were all $9$s. Then the sum of the digits of $5^n$ would be $9\cdot1000000=9000000$, which is far less than the number $2^{1000000}$.  But is there a way to prove this using theory?
Find all $n\in\mathbb N^+$ such that the sum of the digits of $5^n$ is equal to $2^n$.   I've solved this, but I think the proof is a bit weird and I wonder if there's a better one.    My proof:  We can see that $n\neq 1$, $n\neq 2$, $n=3$, $n\neq 4$ and $n\neq 5$.  Let $n&gt;5$. Notice that $5^6$ has $5$ digits, while $\frac{2^6}{9}=7\frac{1}{9}$, so $5^6$ has to have at least $8$ digits in order for the sum of the digits to be able to at least reach $2^6$.  Everytime $1$ is added to the exponent of $5$, the amount of digits $5^n$ gets is at most $1$, while $\frac{2^{n+1}}{9}-\frac{2^n}{9}=\frac{2^n}{9}&gt;3$, so the lowest amount of digits it has to have to reach $2^n$ becomes higher by at least $4$.  When $n=6$, the amount of digits $5^n$ has is already lower than the amount it has to have in order to reach $2^n$, and I've just proved that the amount it has to have in order to reach $2^n$ grows faster than the amount of digits $5^n$ has, therefore $5^n$ will never be able to reach $2^n$ when $n&gt;5$.  Hence the only solution is $n=3$.
Why can't I suggest edits on SE meta sites? When you click &quot;edit&quot; on a meta site as a non-edit-privileged user, the error message appears:  Suggested edits are not allowed on non-tag-wiki posts on meta sites.  Right as I was going to post this as a possible bug, I finally found  which says that per-site Metas don't have suggested edits by-design because it's &quot;not needed&quot;. However, I've been participating a lot over at  where there is a lot still to be figured out in terms of site guidelines.  The Meta is one of the most active I've seen and a lot of the questions are really trying to gather community-built suggestions.  So maybe more of those need to be Community Wiki, but it still brings me to ask why don't per-site-metas &quot;need&quot; suggested edits? Adding bounty: The more betas I participate in, the more I don't get this. Remove note about sudo that appears when opening the terminal Whenever I open the terminal,   To run a command as administrator (user "root"), use "sudo &lt;command&gt;".   See "man sudo_root" for details.   appears before the terminal. How do I get that to go away? We're injecting records into a journey using a Salesforce data event.  I also want to advance the people in the journey based on updates that happen in Salesforce (e.g., as an Opportunity record advances through the sales process), but I can't figure out how to do that.  I've read all the documentation I can find, but there doesn't seem to be a Journey Builder widget that will wait to be prompted by an event in Salesforce before continuing someone on a journey.  At the moment, I'm completely stuck.    I'd greatly appreciate any advice or information on how I can make journeys event-driven in this way.  Thanks very much! I have a Canon Powershot A710 and I previously had a previous Powershot A-series model. I love them due to the high amount of control they offer. Sadly, they both suck in low light conditions, their flash is very unresponsive, and the camera takes several seconds preparing itself for the next shot after a flash shot. And even then, the pictures they take in low light/flash are as bad as good are those with good light.  I'd like a camera which gives a lot of control options, but what do I need to look for to find one that's great in low light, with or without flash? How did Enrico Fermi calculate the classical Fermi Problem? From Wikipedia: Fermi was known for his ability to make good approximate calculations with little or no actual data, hence the name. One example is his estimate of the strength of the atomic bomb detonated at the Trinity test, based on the distance travelled by pieces of paper dropped from his hand during the blast. Fermi's estimate of 10 kilotons of TNT was remarkably close to the now-accepted value of around 20 kilotons, a difference of less than one order of magnitude.  I have not been able to find any references explaining how he made this calculation. Please provide a reference or an example calculation.  See also: I have the amulet of Mara but when I go to talk to maramal about a wedding nothing shows up. WHY is that. Why isn't  generic?    Seems like Collection&lt;E&gt; could have boolean remove(E o);   Then, when you accidentally try to remove (for example) Set&lt;String&gt; instead of each individual String from a Collection&lt;String&gt;, it would be a compile time error instead of a debugging problem later. Resolve build errors due to circular dependency amongst classes How did the AT-ATs get to the surface of Hoth? Pretty new to Blender, I'm trying to learn how to make a studded wall.  How do I use the array modifier to use imperial units? For example, I would like to create a stud, and have them spaced 16 inches on center a set distance.  Thank you much!  Edit I've accepted the below answer, however, I had another issue I had to fix. The culprit was that when I scaled the initial object, I didn't apply the scale. Because I didn't apply the scale before the modifier, it wasn't working as expcted. Thank you much to  for helping me through this issue! ViewExpiredException in Firefox and IE after migrating to Jboss 7.0 AS How to install MariaDB? What is the most stable/safest way to install MariaDB? I'm used to always hearing or seeing a  before certain nouns.  Yet on certain occasions the article is totally omitted, and it bothers me.  I'm wondering what the justification for omitting the article is, especially when the definite article would seem to be required, or if it is just bad English either on the part of the individual in question or more likely by tradition.  The most frequent place I hear this is at the doctor or dentist's office. In every one I have ever visited, the receptionist or assistant always says things like "Doctor will see you shortly" instead of "The doctor will see you shortly."  The wikipedia article linked above says this: "A definite article indicates that its noun is a particular one (or ones) identifiable to the listener".  In the case of a clinical setting, I know who the doctor is, especially if this is not my first visit.  The people in the office know the doctor even better than I, so it would seem to be inappropriate to omit the article. Even the indefinite article would be incorrect. It further says, under "Zero article":  "In languages having a definite article, the lack of an article specifically indicates that the noun is indefinite."  Since the noun is not indefinite, why are they omitting the article?  Another place I see the article omitted is with the word "bar".  Bartenders apparently don't say "I tend the bar" they say "I tend bar".  Unless a bartender works in multiple bars, it would seem like an article would be required.  Furthermore, "the bar" can mean any bar a person goes to, so even in a case like , it seems like the word "the" should be in the sentence.  The only thing I can think of in this case is maybe the speaker doesn't want to confuse the listener into thinking they work in the legal field.  Is this just a bad habit, a historical corruption or tradition, or is there really a valid reason (or perhaps more than one) for these omissions? To get rid of dust that had gathered near the fan, I put a vacuum cleaner to the air intake (with the computer turned off).  This removed the dust.  It also caused the fan to rotate at a fairly high speed. Quite faster than normal speed, judging from the sound.  Do I risk destroying stuff by doing this? How do you handle uneven tiles while rendering a tile map? Here is what I want to do with my tile map that I am unsure of. As you can see the top walls are way larger then the bottom and side ones (this is also an issue with my corners which are odd shapes [more like an L then a square] as well as larger then the 40x40px that my current tiles are).    I have also attached the tile sheet I am using for the tiles to show you what the corners and tops look like compared to the rest of the tiles.  What I’m thinking is I might be able to just draw them using the same array I have if I use the tiles width and height instead of a set WxH? I dont know how well this will work, but were the blocks that will have different values will be blocks the player/NPCs can’t step on then it may not be an issue, again I’m not even sure if that is a viable solution.  My code in case you want to see how I'm currently handling it:  (pastebin) (edit:im not handling it, meant how I'm currently doing the map)  tl;dr - how can I take take my 2d looking tile map and change it to have different size tiles? or is there a better way to do what I am trying to do? Is there a name of a method that I should be looking for while googling? What is a terminal and how do I open and use it? What is a terminal? How do I open a terminal window? How do I use the terminal? All of my Firefox add-ons have been disabled suddenly, how can I re-enable them? A ring element with a left inverse but no right inverse? Can I have a hint on how to construct a ring $A$ such that there are $a, b \in A$ for which $ab = 1$ but $ba \neq 1$, please? It seems that square matrices over a field are out of question because of the determinants, and that implies that no faithful finite-dimensional representation must exist, and my imagination seems to have given up on me :) It is suggested as an exercise in Serge Lang's book "Algebra" to show that the commutator subgroup $G^c$ of a group $G$ is a normal subgroup.   I'd like to do that but I am afraid I need help,   I think the first thing I need to figure out is how a general element in the commutator subgroup looks like, so that I can check that the defining condition for normality is satisfied.  That is, supposing for a moment that a general element in $G^c$ is denoted by $g$, I need to show that $aga^{-1} \in G^c,$ for all $a \in G$.  But here I get stuck, first because I am unsure how to write a general element in $G^c$ - a simple product in $G^c$ is of the form $xyx^{-1}y^{-1}aba^{-1}b^{-1}$ where $a,b \in G$. I cannot see a way to simplify this - I am sure there is one, but somehow I am blind today.  The second thing then is, even if one tries out the conjugation of a simple element like $xyx^{-1}y^{-1}$ in $G^c$, again not simplification offers itself easily I think .. what am I missing ?  An alternative would be to find a homomorphism of $G$ whose kernel is precisely $G^c$ - here I tried to think of this as a map $G \times G \to G$ but whatever I cook up is not a homomorphism.  Thanks for your hints !! I am completely new in Ubuntu. Just installed Ubuntu 14.04. On my other computer running Windows, I am using Avast, free edition.  Do I need anti-virus protection at all?