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Find the largest integer n for which 12^n evenly divides 20!

 Mar 24, 2020
 #1
avatar+1956 
+2

The largest power of 3 that divides 20! is 3^8. 

 

There are 18 powers of 2 and 8 powers of 3 in 20! (prime factorization)

We want the largest value of n such that 2n is less than or equal to 18.

 

That means n is less than or equal to 8, so 8 is our answer.

 

Hope this helped!

 Mar 25, 2020
 #2
avatar+1956 
+2

Is this correct?

CalTheGreat  Mar 25, 2020
 #3
avatar+483 
-1

Cal I do have to ask, in your solution, your rationale was that you want the largest value of n such that 2n is less than or equal to 18. How did you arrive at this conclusion firstly, and secondly, how does that mean n is less than equal to 8? Those 2 don't add up. 18 /2 = 9 not 8, so how did you get that answer?

jfan17  Mar 25, 2020
 #4
avatar+483 
-1

The reasoning here is not explained at all! The reason why n = 8 is because although n would equal 9 for the 22 term, there are only 8 powers of 3 given. Your explanation doesn't make sense in the context of your answer.....

jfan17  Mar 25, 2020
 #5
avatar+1956 
+1

I'm sorry my explanation doesn't make sense.......

 

Okay. Here is my edited one:

 

The largest power of 2 that divides 20! is 2^18 .

The largest power of 3 that divides 20! is 3^8. 

We want the largest value of n that n is less than or equal to 8.

 

Therefore, 8 is our answer.

 

Does this make sense, jfan?

 Mar 25, 2020
 #6
avatar+483 
0

No? That doesn't explain anything. Please understand your solutions before you post them! You're still not explaining WHY you have n less than or equal to 8...... if I was talking from an uneducated perspective, I would have NO idea with your logic jumps like that

jfan17  Mar 25, 2020
 #7
avatar+1956 
+1

I really have nothing to say to that.....:(

CalTheGreat  Mar 25, 2020
edited by CalTheGreat  Mar 25, 2020
 #9
avatar+1904 
0

Well I have something to say... jafan17, if you were talking from an uneducated perspective, you would not have noticed the logical fallacy. However, if you were speaking from an educated perspective, but ignorant of the answer then the logical fallacy would be apparent.     

 

There are many solutions on this forum (and elsewhere) presented from a perspective that requires knowing the answer to the question to create the logical steps to the answer.   I sometimes refer to these as Math Magic, Pixie Dust solutions. There is a guest member (Mr. BB) who is notorious for using Math Magic and Pixie Dust logic as a solution process to justify his answers.  They are worthless, time-wasters for students needing to learn the concepts and process for solving equations and deriving answers. 

 

An easy method to confirm the invalidity of such logic is to solve a similar question using this derived logic. If the method does not lead to a solution, then the logic is invalid and untenable.   

 

Learning mathematics is usually a gradual process. With students, first, leaning the rote mechanics, and then the logic behind the process. The mantra of “ours is not to reason why, but to invert the divisor and multiply” is mostly universal up to the higher levels of mathematics. 

----------

 

jafan17, now that you’ve ripped Cal a new one for her incompetence, you could demonstrate the correct, logical process, to educate Cal and others who will read this in the future.  A simplified explanation of Legendre’s formula would probably be the optimal choice. 

 

 

GA

GingerAle  Mar 25, 2020
 #10
avatar+483 
0

-GA

 

First of all, if I was really talking from an unknowledgeable perspective, what is your rationale behind me not noticing the logical fallacy presented by Cal? This makes no sense whatsoever. Cal tells us that :

 

"

The largest power of 2 that divides 20! is 2^18 .

The largest power of 3 that divides 20! is 3^8. 

We want the largest value of n that n is less than or equal to 8.

"

If I really had no knowledge of Legendre's formula like you so eagerly recommended, how would I have known anything about this kind of reasoning? If I was really uneducated and was looking at the steps Cal used as opposed to just the answer, I would've said "wait, how does this logic follow through"? Because if I didn't know anything about the subject, then I wouldn't have understood those last two lines at all. Your assumption here is just blatantly false.

 

Secondly, sure, say I buy your argument that there are many solutions on this forum presented from a knowledgeable perspective. Does that mean we should present all of our solutions from "a perspective that requires knowing the answer"? No, it does not by any means make us assume that our question-askers know what we're talking about. Rather, we should go through each step and explain the logic behind it. I'd argue that this is comparatively better than making them "learn the rote mechanics, and then the logic behind the process" themselves as you advocate for, because when we show question-askers the logic and decision-making calculus behind our steps, that allows them to go forward with more knowledge to tackle different types of these problems, which they can always combine with a bit of self-learning. We shouldn't just post the answer with logical leaps and expect our readers to understand. 

 

Thirdly, it was a mistake on my part to not fully explain Legendre's as you mentioned, however, I'm not trying to rip "Cal a new one for her incompetence", I'm trying to help her so she doesn't post misleading solutions that she doesn't fully understand; that just causes confusion for the question askers and everyone in general. 

jfan17  Mar 25, 2020
 #14
avatar+1904 
0

Jfan17 wrote:

First of all, if I was really talking from an unknowledgeable perspective, what is your rationale behind me not noticing the logical fallacy presented by Cal? This makes no sense whatsoever.  ... ... ... Your assumption here is just blatantly false.

 

Of course, it makes no sense whatsoever; I have no rationale, because that is not what I said.

 

You DID notice the logical fallacy; however, you could not have noticed it from an uneducated perspective or unknowledgeable perspective. (In your current post, you use “unknowledgeable perspective,” in your original post you use “uneducated perspective.” I assume you intend the same meaning for unknowledgeable and uneducated.)  I noticed that you noticed it, so my assumption cannot be blatantly false. You implied that Cal should explain this as if your observation point was from an uneducated perspective.  

 

Consider some analogous examples:

“Cal, I’m adopting the perspective of a blind person, now explain to me the colors of a rainbow.”

“Cal, I’m adopting the perspective of a deaf person, now explain to me the decrepitating sounds of a fart.

“Cal, I’m adopting the perspective of an amoeba, now explain to me the meaning of Shakespeare’s Sonnet 116, ‘Let me not to the marriage of true minds Admit impediments.’”

 

Like your “uneducated perspective,” these perspectives are more suited for Zen philosophy than science or math.   

 

If I really had no knowledge of Legendre's formula like you so eagerly recommended, how would I have known anything about this kind of reasoning?

 

You wouldn’t have ... and that was my point.  That is not doable. However, you could be ignorant of the answer to this problem, and still know that Cal’s logic is fallacious or correct.

 

----------------------

 

Secondly, sure, say I buy your argument that there are many solutions on this forum presented from a knowledgeable perspective. Does that mean we should present all of our solutions from "a perspective that requires knowing the answer"?

 

OK... I’m seeing a pattern here. It seems that some of the nuances of the English language are escaping your attention.  There is a big difference between a knowledgeable perspective and a perspective that requires knowing the answer to the question.  Maybe you are unknowledgeable of these nuances, but you don’t seem uneducated...

 

I wrote, There are many solutions on this forum (and elsewhere) presented from a perspective that requires knowing the answer to the question to create the logical steps to the answer.  I sometimes refer to these as Math Magic, Pixie Dust solutions. 

 

These Math Magic, Pixie Dust solutions only appear to work.  The answer may be correct but solution gives only an illusion that the answerer solved the problem.

 

This forum has more than few answers like this. Here are some examples.

 

Here’s a Fluff and Blarney example.  https://web2.0calc.com/questions/nice-question#r6  Melody says this is “intuition.”  There may be some intuition in this, but it is so disordered and chaotic it is functionally the same as a Math Magic, Pixie Dust solution. Mr. BB(1) is notorious for these Blarney solutions; he continues to use them, though I’ve trolled him and ripped him a new one, many times over the years.  Here’s a recent example.

 

Here’s an example of a Magic, Pixie Dust solution for a math question that isn’t a math question –it’s just BS.  https://web2.0calc.com/questions/easy-but-hard-question#r27  The question and “answer” originated from a YouTube channel. The “solution” and it reasoning is consistent with something any of the BB brothers might use.

 

Here’s a student visiting the forum who is actually looking for “magical solutions” for her question. Specifically, the student wants the logical process of the solution to convey to her an understanding of the required prerequisites –without the required study, of course.  

https://web2.0calc.com/questions/statistics-please-help-ive-gotten-different-answers#r2

 

Here I troll a Mr. BB-like observation that uses Pixie Dust to give confidence, credibility, and priority to a personal expectation rather than a mathematical expectation.  https://web2.0calc.com/questions/two-standard-dice-are-rolled-what-is-the-expected#r3

 

In this post, I use my cat’s telekinetic powers to troll Mr. BB for his analysis using magical incantations and charmshttps://web2.0calc.com/questions/rolling-dice#r14

 

This one is quite funny...  A troll post full of metaphors: Generating functions that seem magical, magical detections of unique patterns, and a god with the power to change mathematical and physical constants.  https://web2.0calc.com/questions/combinations-and-permutations_5#r6

---------

 

... we should go through each step and explain the logic behind it. I'd argue that this is comparatively better than making them "learn the rote mechanics, and then the logic behind the process" themselves as you advocate for, because when we show question-askers the logic and decision-making calculus behind our steps, that allows them to go forward with more knowledge to tackle different types of these problems, which they can always combine with a bit of self-learning. We shouldn't just post the answer with logical leaps and expect our readers to understand.


Rote learning is an optimal first step in structured learning, and it is a time-honored method for teaching young minds. It’s natural for most minds to develop at least a minimum logical reasoning behind the process, and some will completely figure out the logic. That is, they go forward, anyway.  In any case, it is much easier to teach a mind the abstract logic behind a process after its exposure to the rote mechanics.  To dumb-it-downed with an analogy, a wet sponge absorbs water much more easily than a dry sponge.   

 

Consider a non-human mind –a chimp. This chimp uses a rock to smash hard nuts, and slobbers on a twig to fish for termites in a terminarium . This chimp probably learned this from another chimp.  I doubt the teaching chimp related the classical, Newtonian physics, mass-energy equations of Gottfried Leibniz, Johann Bernoulli, and Lord Kelvin, before sending the chimp out to crack nuts. Nor is it likely that the student chimp received instruction on the use of Young-Dupré adhesion equations before going out to fish for termites.

 

[However, we Genetically Enhanced Chimps, as a matter of course, teach these equations to our infants. They are expected to learn these, and much more before five-years of age, else they are sent off to join a troop of non-genetically enhanced Chimps. ]

--------------

 

Thirdly, it was a mistake on my part to not fully explain Legendre's as you mentioned, however, I'm not trying to rip "Cal a new one for her incompetence", I'm trying to help her so she doesn't post misleading solutions that she doesn't fully understand; that just causes confusion for the question askers and everyone in general.

 

That’s excellent. [That’s one of the main reasons why I troll someone. As for ripping them a new one, I used to be a proctologist in a previous life, and old habits die hard.] 

 

We Genetically Enhanced Chimps also have an instinct to give cautionary warnings to our children to “be careful” while playing. When children are playing with math and science, a general warning should always be “Do not embrace the philosophies of the brain-dead and intellectually disturbed.  It’s OK to entertain yourself and others with this drivel, but embracing it will alter your perception of reality and make it easier to accept the next cup-full of drivel.”

 

I’ll add that probably Cal will eventually learn the logic behind the math. She’s eleven, so there is more than enough time for progress.  Also, considering her age, Cal is rather a wonder to behold, in comparison to the notably older, moronic junior and senior high school students that are now the forum’s collective centerpiece of counterintelligence, relentlessly mooching for homework answers for often brain-dead and repetitive questions.     

 

Even so, Cal can definitely cause confusion for the question askers and everyone in general, but the magnitude of her effect is relatively minimal.  While there should never be a free pass for anyone crossing the Troll Bridge, a discounted pass for academics, (not plagiarism) for student youth may be an acceptable compromise.  

 

By contrast, the BBs are (senior) adults and relentless screw-ups. I doubt the BBs will ever learn, and I hope they never do. The BBs are the bane of web2.0calc and that makes them great and worthy troll bait –a stable ancient staple, in perpetual disparity to the other screw-ups on the forum who come and go.  

 

Mr. BB, the stubborn, relentless, intractable Blarney Banker of lore: a pseudo intellectual with a multiplicity of advanced dimwit degrees in arrogant stupidity; a professor of misinformation, who teaches with authority and irritation. ...

https://web2.0calc.com/questions/kim-has-10-identical-lamps-and-3-identical-tables#r11

 

-----

With all the new brats on here, this forum needs another troll, even more than it needs another mathematician. You seem well educated, skilled in basic mathematics, with good communication skills, and you have natural trolling talent. You have all the ingredients to be an excellent troll on here.  I hope you continue to use all of your talents on this forum. 

 

GA

GingerAle  Mar 29, 2020
edited by GingerAle  Mar 29, 2020
 #15
avatar
0

ginger ale hanging with kids

Guest Mar 29, 2020
 #16
avatar+632 
0

I love the age gap. Here we have an elementary schooler, a highschooler, and who knows how old GingerAle is. It's almost like a food chain :/

 #8
avatar+111321 
+2

Maybe not  the most elegant way to do this....but  .....

 

20!  =

 

20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2

 

(2 * 2 * 5) (19) ( 3 * 3 * 2) * 17 * (2 * 2 * 2 * 2) * (3 * 5)  * ( 7 * 2) * (13) * (12) * (11) * (2 *5)  * (3 * 3) * (2 * 2 * 2) * (7) *

(2 * 3)  * ( 5) * ( 2 * 2) * ( 3 * 2)

 

In addition to 12, we only need consider  powers  of 2  and  3  

 

[ ( 2 * 2 )  ( 3 * 3 * 2 ) ( 2 * 2 * 2 * 2)  (3 * 2 * 12 * 2 ) (3 * 3 * 2 * 2*  2) ( 2 * 3 * 2 * 2 * 3 * 2) ]  =

 

[ 2 ^16  *  3^7  * 12 ]     { Break  this up into  powers of 12 }    

 

[ 3 * 2^2 ]  [ 3 * 2^2 ] [ 3 * 2^2 ]  [ 3 * 2^2 ] [3 *2^2 ] [3 * 2^2]  [ 3 * 2^2 ]  * 12   *   [ 2^2] ignore this

 

     12  *      12     *     12     *     12     *    12   *    12    *    12     *   12

 

12^8   will  divide 20!

 

So

 

n =  8

 

 

cool cool cool

 Mar 25, 2020
edited by CPhill  Mar 25, 2020
 #13
avatar+1956 
+1

Thanks, Chris. That is a really good answer.... kind of what I was trying to say, but couldn't. 

 

Anyway, I've learned something, and I'm happy about it.

 

If it will help your argument, I will leave.

CalTheGreat  Mar 25, 2020

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