How Can We Make 100! Divisible by 12^{49}?

[OmCheeto]

Care to explain?

I wouldn't expect Citan to explain, as this looks very similar to the https://www.physicsforums.com/showthread.php?t=450364" thread.

Citan is correct, point 2 is false. And I'm not even a mathematician. As a matter of fact, I cannot understand a single thing in this entire thread, except that point 2 is false.

 

[Niivram]

Yeah, I'm convinced that Point 2 is wrong. But I'm still confused. You're saying, if you have an infinite number of small circles, and an infinite number of big circles, the area isn't the same right?

 

[OmCheeto]

Yeah, I'm convinced that Point 2 is wrong. But I'm still confused. You're saying, if you have an infinite number of small circles, and an infinite number of big circles, the area isn't the same right?

As I said, I am not a mathematician, and got way over my head linguistically in the troll physics thread. I will not make the same mistake again.

I also believe you've broken the rules of the thread:

A Q&A game is simple: One person asks a relevant question (it can be research, calculation, a curiosity, something off-the-top-of-the-head, anything ... as long as it's a math question) and other people try to answer. The person who posts the first correct answer (as recognized by s/he who asked the question) gets to ask the next question, and so on.

Let me start this off with a simple number theory problem :

What is the least number than must be multiplied to 100! (that's a factorial) to make it divisible by 12^{49} ?

(throw in a brief -couple of lines or so- explanation with the answer)

You need to answer the last question before you can ask your own.

And I'm afraid I'm going to have to unsubscribe from this thread. I read the post(#247) before your original, and can't figure out if it's a question or an answer.

Ciao!

 

[Niivram]

hehe, Never thought of the rule : D. Sorry.

Ciao!.

 

[Erland]

O well since I guess I got the last one, I'll just ask again what post 169 asked: How can one find the surface area of a ring?

What do you mean by a "ring"? A torus?

 

[Gib Z]

I may have, but I have a better question now.

If a_n is a monotone decreasing sequence such that \sum a_n converges, show na_n \to 0, and then make the generalization that a measure theorist would make.

 

[Erland]

I may have, but I have a better question now.

If a_n is a monotone decreasing sequence such that \sum a_n converges, show na_n \to 0, and then make the generalization that a measure theorist would make.

Well, the proof is quite easy: We must have a_n\ge 0 for all n, for otherwise we would have a series in which all but finitely many terms are smaller than some fix negative number, and then the series would diverge to -\infty. Assume, to get a contradiction, that na_n \to 0 does not hold. This means that there is an \epsilon>0 and arbitrary large m such that ma_m\ge\epsilon.
Now, there is an N such that \sum_{n=N+1}^\infty a_n <\epsilon/2. Then, we can choose an m\ge 2N such that ma_m\ge\epsilon. Since a_n is monotone decreasing, it folllows that a_n\ge\epsilon/m for all n\le m. Hence, \sum_{n=N+1}^m a_n\ge (m-N)\epsilon/m \ge\epsilon/2, which is a contradiction. Thus, na_n \to 0.

Generalization? The only one that comes to my mind is: Let \mu be a positive measure on the real interval [a,\infty), ant let f be a \mu-integrable on [a,\infty) (i.e. the integral over the entire interval is exists and is finite) and monotone decreasing. Then xf(x)\to 0 as x\to\infty.

 

[Number Nine]

Care to explain?

The sum of the areas of the circles in figure 1 is bounded by the area of the triangle containing them, which is obviously far less than the area of the central circle in figure 3. You could probably show that the areas of the triangle in fig 1 and the squares in figs 2 and 3 are the least upper bounds for the summed areas of the circles in those respective figures, thus...

fig 1 < fig 2 < fig 3

 

[Inventor 4U2]

I got a challenge for you... This question is answered in two different ways from two different professionals with different backgrounds. I asked a Physics Professor and a Mathematics Professor a question and got two different answers. But, isn't it true there is only one truth in answering a simple question such as this one? Here it is- if I take a distance or an object and cut it perfectly and half, then take one of those halves and cut it perfectly in half, again. And repeat this over and over again. What will happen eventually to the distance or thickness of the object ? The reason I brought this to your attention is because it appears that you pride yourself and/or you have a good understanding of physics, I am assuming. Good luck, I would love to hear your response to this question.

 

[Erland]

Inventor, what do you mean with "eventually"? Do you mean that you actually cut an infinite number of times, or just that you cut a finite, but arbitrarily large number of times?

 

[Number Nine]

I got a challenge for you... This question is answered in two different ways from two different professionals with different backgrounds. I asked a Physics Professor and a Mathematics Professor a question and got two different answers. But, isn't it true there is only one truth in answering a simple question such as this one? Here it is- if I take a distance or an object and cut it perfectly and half, then take one of those halves and cut it perfectly in half, again. And repeat this over and over again. What will happen eventually to the distance or thickness of the object ? The reason I brought this to your attention is because it appears that you pride yourself and/or you have a good understanding of physics, I am assuming. Good luck, I would love to hear your response to this question.

Can you elaborate? The question is very unclear, which may explain why you received two different answers. What do you mean by "what will happen to the thickness of the object"? Are you asking what will happen if you halve its length an infinite number of times?

 

[TheMarksman]

I got a challenge for you... This question is answered in two different ways from two different professionals with different backgrounds. I asked a Physics Professor and a Mathematics Professor a question and got two different answers. But, isn't it true there is only one truth in answering a simple question such as this one? Here it is- if I take a distance or an object and cut it perfectly and half, then take one of those halves and cut it perfectly in half, again. And repeat this over and over again. What will happen eventually to the distance or thickness of the object ? The reason I brought this to your attention is because it appears that you pride yourself and/or you have a good understanding of physics, I am assuming. Good luck, I would love to hear your response to this question.

nothing, it remains the same..it`s a question similar to the question "which is heavier?" a kilo of cotton or a kilo of nails..lol

 

[pwsnafu]

But, isn't it true there is only one truth in answering a simple question such as this one?

Definitely not. Most "simple questions" tend to be the hardest because the terms used are open to interpretation.

if I take a distance or an object and cut it perfectly and half, then take one of those halves and cut it perfectly in half, again. And repeat this over and over again. What will happen eventually to the distance or thickness of the object ?

A more interesting problem would be catalog as many possible answers as we can think of!

1. An infinite number of divisions of space ("distance") is possible, and the limit is zero.
2. You are dividing space, but don't have infinite time, so it is impossible.
3. You are dividing space, and have infinite time, but you can't divide smaller than Plank's distance.
4. You take shorter time with each subdivision and you can divide Plank's distance, but you can't observe it. So it is moot.
5. You are dividing a physical object. You can't divide subatomic particles.
6. Heck, you can't divide an atom perfectly either.
7. You can't do perfect division, full stop. The question is meaningless.

 

[Inventor 4U2]

Hello "pwsnafu". I see that you pretty much understood what I was asking. I was asking the question using two different scenarios. Cutting a distance in halves until there's nothing left, if possible OR cutting on object in halves until there is nothing left, if possible. Well, you explained it well that by cutting a PHYSICAL object in half over and over again you will eventually get down to subatomic particles which apparently are impossible to cut in half, again physically. I also agree with you when you said "An infinite number of divisions of space is possible, and the limit is zero.". I agree with you 100%. But, why is it so difficult for so many other people including so-called professionals to answer the question properly when I was referring to DISTANCE being cut in half over and over and over and over again, without putting a limit on how many times you can do it over and over. I had to word it that way and if I worded it differently such as saying "over and over, what will happen eventually" it leads people to assume there is an end conclusion which leads them to say "you will run out of space", which I know is absolutely incorrect. That is why I asked the question the way I did but there may have been a better way to ask it. The two persons whom left the other posts above yours agree with that, apparently. It was at Davis, California University where I had asked a Physics and Mathematics Professor the question. And when I asked the two professors the question I used ONLY the scenario using a space or distance and NOT on object. The physics professor responded to my question by saying "Yes, you will run out of space and the two objects will touch each other." Why did he answer the question that way, word for word. Keep in mind I asked if you have two objects and referring to the distance between the two objects that are coming closer to each other. I could not have been more specific to what I was referring to when I asked that Davis University Physics professor. But, I had waited about one half hour for the mathematics professor to return to his class on the campus to ask HIM the exact same question. He responded and said "The two objects will never touch each other no matter how many times you cut the distance between the two objects in half." I would love to see both of those professionals in the same room and at the same time discussing their reasoning for reaching their conclusion to each other! Again, believe me, I did ask both those professors the exact same question using the exact same words. I already knew the answer to my question but, I'm just trying to make some sense out of why I got two completely different answers. I guess sometimes the smarter we think we are, that is assume we are, the more sloppy we get at answering the most basic simple questions that are somewhat easy to answer. With all the above being said and realized, I guess there is another way to share the thought of what happens when you cut a space between two objects over and over continuously and here it is: "There is a way to prove how you can take two objects and make them move toward each other for an eternity but they will never touch each other!" It's called MATHEMATICS. Meaning it can be proved mathematically but NOT physically.

 

[Number Nine]

The responded differently because they understood the question differently. They had to speculate as to what you meant just like we did, and since they speculated differently, they arrived at different answers.

 

[Inventor 4U2]

Thank you for your comment regarding my choice of words, "eventually". You and others were correct on wondering what I was referring to when I chose to use the word "eventually" when asking the question what happens eventually when you keep cutting a distance between two objects in half over and over again. I assumed my word vocabulary was better than that. The word eventually does not mean to infinity. Rather it is a undetermined amount of time. Oops! Sorry about that.
I've been out of town for four weeks searching for meteorites but ended up finding historic World War II 50 caliber and 20 mm bullets and shells on the desert in Arizona. I would've responded sooner to your post. Thanks and no reply necessary.

 

[LCKurtz]

This reminds me of this puzzle: An attractive woman is waiting for a male mathematician who is 10 meters away. As each minute passes, the man moves half the remaining distance closer.

Question: Will the man reach the location woman in any finite time?
Answer: No, but he will get close enough for all practical purposes.

 

[Number Nine]

This reminds me of this puzzle: An attractive woman is waiting for a male mathematician who is 10 meters away. As each minute passes, the man moves half the remaining distance closer.

Question: Will the man reach the location woman in any finite time?
Answer: No, but he will get close enough for all practical purposes.

An infinite number of mathematicians walk into a bar; the first orders a glass of beer, the second orders half a glass of beer, the third orders a fourth of a glass of beer...

The bartender says "You're all idiots!" and pours two glasses of beer.

 

[DaveC426913]

An infinite number of mathematicians walk into a bar; the first orders a glass of beer, the second orders half a glass of beer, the third orders a fourth of a glass of beer...

The bartender says "You're all idiots!" and pours two glasses of beer.

 

[Jimmy Snyder]

How many Lebesgue measurable subsets of the reals are there?

 

[Deveno]

How many Lebesgue measurable subsets of the reals are there?

i tried counting them, but i gave up after aleph-null...

 

[pwsnafu]

How many Lebesgue measurable subsets of the reals are there?

I just came across this:

Let C be the Cantor set and E = C \times [0,1]^{k-1} \subset R^k. Then E is uncountable with cardinality c and with Lebesgue measure zero. So there are 2^c subsets of E, each Lebesgue measurable.

 

[Jimmy Snyder]

I just came across this:

Let C be the Cantor set and E = C \times [0,1]^{k-1} \subset R^k. Then E is uncountable with cardinality c and with Lebesgue measure zero. So there are 2^c subsets of E, each Lebesgue measurable.

This is close, but I am looking for subsets of the reals. So, set k to 1 and you have shown that lower limit is at least 2^c. Now just provide an upper limit.

 

[LCKurtz]

This is close, but I am looking for subsets of the reals. So, set k to 1 and you have shown that lower limit is at least 2^c. Now just provide an upper limit.

But isn't the cardinality of all subsets of the reals $2^c$, so that is also an upper limit?

 

[Jimmy Snyder]

But isn't the cardinality of all subsets of the reals $2^c$, so that is also an upper limit?

Yes, you have solved it.

 

[godsaveme]

Topic:

Two birds in the tree, the hunter shot one.

Ask:

Only a few were left in the tree? Live or die?

You need to determine the answer

 

[DaveC426913]

Only a few were left in the tree? Live or die?

What??

Sorry, I cannot parse those sentence fragments.

 

[godsaveme]

What??

Sorry, I cannot parse those sentence fragments.

I am a chinese,my english is poor.
That may be how many birds in the tree?

 

[godsaveme]

This is a certainty and uncertainty question!

 

[godsaveme]

How to determine?
The number of birds，live or die？

 

[godsaveme]

Can you help me to translate chinese into english?

 

[Citan Uzuki]

Zero. The other bird flew away.

Edit: Or maybe one, if the bullet didn't knock the first bird off the perch. In any case, the live bird is gone.

 

[jgutierrez218]

I don't think you can answer questions like that.

There are two birds to begin with, one is shot dead. 1 is left.

1 does not equal a few, as a few is generally equated to mean 5.

 

[Citan Uzuki]

Actually I can answer questions like that. Proof: I just did

I think the "a few" was just a translation failure on the part of our Chinese friend.

 

[Jimmy Snyder]

I heard this one before.
He: There were two birds in the yard and I shot one of them. How many were left in the yard?
She: One.
He: No, one. The one that I shot. The other one flew away.

 

[Citan Uzuki]

Okay, guess it's my turn to ask a new one.

What are all the continuous functions f:\mathbb{C} \rightarrow \mathbb{C} such that \forall z,w\in \mathbb{C},\ f(z+w) = f(z)f(w)? Does the answer change if continuous is replaced with measurable?

 

[DaveC426913]

...a few is generally equated to mean 5.

What?? Where did you get this?

For me, "a few" is three or more.

 

[Jimmy Snyder]

2 is a couple. 3 is a crowd. 3 to 7 is a few. 5 to 10 is some. 8 to 15 is several. 15 to 37 is a bunch or if it is something you don't like, then it's many, or even too many if you really don't like it. 30 - 100 is a profusion. 100 - 1000 is a multitude. More than that is a plethora or a surfeit.

 

[Mandlebra]

2 is a couple. 3 is a crowd. 3 to 7 is a few. 5 to 10 is some. 8 to 15 is several. 15 to 37 is a bunch or if it is something you don't like, then it's many, or even too many if you really don't like it. 30 - 100 is a profusion. 100 - 1000 is a multitude. More than that is a plethora or a surfeit.

Oh yes, so often do I ask for a crowd of things.

 

[20Tauri]

I don't think you can answer questions like that.

There are two birds to begin with, one is shot dead. 1 is left.

1 does not equal a few, as a few is generally equated to mean 5.

Hahaha, have you guys seen the xkcd strip about this sort of thing?

More seriously, I can see "one" being a valid value for "a few," although this is certainly not its most common usage. It would sort of be analogous to the way "some" is used to mean "at least one" in formal logic.

 

[dkotschessaa]

Ooh, can we bring this back? We did a math trivia type game in math club and I have a few good ones, ranging from basic high school algebra, through analysis and some historical trivia.

 

[physics kiddy]

Let me see if I can do ...

Once we have figured out that 100! has 2^97 * 3^48 in it. Factorise 12^19.
It's (2*3*2)^49 = 2^98*3^49. So the number is 2*3 = 6...