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

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In summary, the conversation was about a Q&A game where one person asks a math question and others try to answer it. The first correct answer gets to ask the next question. One of the questions was about finding the least number that must be multiplied to 100! to make it divisible by 12^{49}. The correct answer was 12^{49} / 100!. There was also a question about why mathematicians often forget to specify that they require a whole number solution.
  • #281


Can you help me to translate chinese into english?
 
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  • #282


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.
 
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  • #283


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.
 
  • #284


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

I think the "a few" was just a translation failure on the part of our Chinese friend.
 
  • #285


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.
 
  • #286


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

What are all the continuous functions [itex]f:\mathbb{C} \rightarrow \mathbb{C}[/itex] such that [itex]\forall z,w\in \mathbb{C},\ f(z+w) = f(z)f(w)[/itex]? Does the answer change if continuous is replaced with measurable?
 
  • #287


jgutierrez218 said:
...a few is generally equated to mean 5.

What?? Where did you get this?

For me, "a few" is three or more.
 
  • #288


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.
 
  • #289


Jimmy Snyder said:
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.
 
  • #290


jgutierrez218 said:
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.
 
  • #291


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.
 
  • #292


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...
 
<h2>1. How do we make 100! divisible by 12<sup>49</sup>?</h2><p>To make 100! divisible by 12<sup>49</sup>, we need to find the highest power of 12 that is a factor of 100!. This can be found by dividing 100 by 12, which gives us 8. This means that 12<sup>8</sup> is the highest power of 12 that is a factor of 100!. To make it divisible by 12<sup>49</sup>, we need to multiply 12<sup>41</sup> to 100!.</p><h2>2. Can we make 100! divisible by 12<sup>49</sup> without changing its value?</h2><p>Yes, we can make 100! divisible by 12<sup>49</sup> without changing its value by multiplying it with 12<sup>41</sup>. This ensures that the value of 100! remains the same, but it becomes divisible by 12<sup>49</sup>.</p><h2>3. Why is it important to make 100! divisible by 12<sup>49</sup>?</h2><p>Making 100! divisible by 12<sup>49</sup> is important because it allows us to easily perform calculations involving large numbers. By making it divisible by 12<sup>49</sup>, we can break down the calculation into smaller, more manageable parts.</p><h2>4. What is the significance of 12<sup>49</sup> in this context?</h2><p>12<sup>49</sup> is the highest power of 12 that is a factor of 100!. This means that it is the smallest number that can divide 100! without leaving a remainder. It is significant because it allows us to make 100! divisible by a large number, making calculations easier.</p><h2>5. Is there a general rule for making a factorial divisible by a large number?</h2><p>Yes, there is a general rule for making a factorial divisible by a large number. We need to find the highest power of the number that is a factor of the factorial, and then multiply it to the factorial. This will ensure that the factorial is divisible by the large number without changing its value.</p>

1. How do we make 100! divisible by 1249?

To make 100! divisible by 1249, we need to find the highest power of 12 that is a factor of 100!. This can be found by dividing 100 by 12, which gives us 8. This means that 128 is the highest power of 12 that is a factor of 100!. To make it divisible by 1249, we need to multiply 1241 to 100!.

2. Can we make 100! divisible by 1249 without changing its value?

Yes, we can make 100! divisible by 1249 without changing its value by multiplying it with 1241. This ensures that the value of 100! remains the same, but it becomes divisible by 1249.

3. Why is it important to make 100! divisible by 1249?

Making 100! divisible by 1249 is important because it allows us to easily perform calculations involving large numbers. By making it divisible by 1249, we can break down the calculation into smaller, more manageable parts.

4. What is the significance of 1249 in this context?

1249 is the highest power of 12 that is a factor of 100!. This means that it is the smallest number that can divide 100! without leaving a remainder. It is significant because it allows us to make 100! divisible by a large number, making calculations easier.

5. Is there a general rule for making a factorial divisible by a large number?

Yes, there is a general rule for making a factorial divisible by a large number. We need to find the highest power of the number that is a factor of the factorial, and then multiply it to the factorial. This will ensure that the factorial is divisible by the large number without changing its value.

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