Pls help with this math problem

G

GalaxyGV

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Mar 27, 2020
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In school I go the following question.
There are 15 composite numbers, each not exceeding 2020. Prove that there are two numbers among them with their common divisor greater than 1.
Please help
 
What have you tried? Where are you stuck? Please tell us so we can help you! :)
 
It's just that i understand that there are a lot more composite numbers than 15 so i am not sure what the question is asking of me
 
Thank you for posting back so fast!

A composite number is any number that has another factor besides itself and 1. Can you list some of these?
 
There you go! Now you see that there are a lot more than 15. The problem does not state that there are 15 composite numbers under 2020, you just have to prove that 2 of these have a common divisor of at least 2.

Can you proceed?
 
Suppose all 15 had no common divisors.

Then the factors of these 15 numbers, at best, are the first 30 primes.

Can you construct 15 numbers all under 2020 using the 2 factors each from the first 30 primes? (I bet you can't)
 
So that means that I could literally just take the first 15 even numbers excluding 2 and they would all have the common divisor as 2
-I think I understand thanks a lot!
 
GalaxyGV said:
It's just that i understand that there are a lot more composite numbers than 15 so i am not sure what the question is asking of me
Click to expand...
Have a look at this discussion of composite numbers .
Yours is a pigeonhole question. In any collection of composite fifteen numbers less \(2020\) two will have a common divisor greater than one. i.e. they will not be relatively prime.
 
I have a problem with the working of this problem. There are 15 composite numbers-Does that mean there are a special list of 15 composite numbers or does it mean any 15 composite numbers?
 
What are the prime numbers less than sqrt(2020)? Why does this help? Combine this with pka's hint of using the pigeonhole theorem.
You really do not need anything else.
 
GalaxyGV said:
So that means that I could literally just take the first 15 even numbers excluding 2 and they would all have the common divisor as 2
-I think I understand thanks a lot!
Click to expand...
No. The problem should state any set of 15 composite numbers no exceeding 2020....
So you can't just pick the 15 which you like. They need to be any 15--you will not know which 15 you are using because then they will be a specific 15 not just any 15. This is what math is all about, proving some properties of numbers without even know the list of numbers.
 
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