[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 307

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Mathemat >>:
Можно ли вычеркнуть менее 43 чисел?

it is possible. For example, return any two prime whose product is greater than 44, say 41 and 43, and cross out their product itself 1763. If we try to return at least one more prime, e.g. 37, then we will have to cross out two more - 1517 and 1591, i.e. the minimum number, apparently, 42



 
Alsu, you forgot about squares 41 and 43. They should be crossed out too.
The condition of the problem "two others of the remainder" implies "different from the product", but not necessarily "different".
The answer in the textbook is 43.
Shall we try to prove it - or is it the solution?
 
Mathemat писал(а) >>
Alsu, you forgot about squares 41 and 43. You should cross them out too.
The condition of the problem "two others of the remainder" implies "different from the product", but not necessarily "different".
The answer in the textbook is 43.
Shall we try to prove it - or is it the solution?


As far as I understand it, the numbers in that sequence are different. Consequently, there are no 2 identical, i.e. no need to cross out the squares, just on the grounds that they are squares.

 
alsu писал(а) >>

it is possible. For example, return any two prime whose product is greater than 44, say 41 and 43, and cross out their product itself 1763. If we try to return at least one more prime, for example 37, we should cross out 2 more - 1517 and 1591, i.e. minimal number, probably, 42


You are wrong.
43 * 45 = 1935
43 * 46 = 1978
41 * 45 = 1845
41 * 46 = 1886
41 * 47 = 1927
41 * 48 = 1968

That is, 41 and 43 have to be crossed out: 1763, 1845, 1886, 1927, 1935, 1968, 1978

 
I.e. by returning 41 and 43 you have to cross out: 1763, 1845, 1886, 1927, 1935, 1968, 1978<br / translate="no">.
PapaYozh, yeah, I didn't notice that myself :)
As far as I understand, the numbers in that sequence are different. Hence no 2 are the same there, i.e. no need to cross out the squares, just on the basis that they are squares.
No, not different, but different from the work. It's something else. I.e. 43*43 = 1849 is perfectly legal, but 1849*1 = 1849 is not.
 
Mathemat писал(а) >>
No, it's not different, it's different from the piece. It's something different. I.e. 43*43 = 1849 is perfectly legitimate, but 1849*1 = 1849 is not.

There we are talking about "set of numbers" and "product of two numbers". It seemed to me that they are talking about different numbers, otherwise the set becomes infinite.
In principle, it doesn't matter. The important thing is that you should remove all numbers from 2 to 44, as it was stated at once. There is no way to remove less.

 
PapaYozh, what about the proof?
What if it is possible to cross out 42 numbers in some perverse way - not necessarily from the beginning of a natural series?
 
Mathemat писал(а) >>
PapaYozh, what about the proof?
What if you can cross out 42 numbers in some perverse way - not necessarily from the beginning of a natural series?


The smaller the number, the more products it can participate in. So it's more efficient to cross out numbers from the beginning of the sequence. There is no point in crossing out "1", that's what you wrote about.

 
OK, here's the solution to the problem of crossing out:

Yes, the solution is not very complete, to say the least. There is no mention of perversions.
Next, the promised one (8th):
 
№337
== 100
1...300301302303304305306307308309310311312313314...628
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