Need an explanation for this phenomenon about LCM's

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Anonymous
Need an explanation for this phenomenon about LCM's 2017-01-15 04:50:11 Post No. 8603283
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Need an explanation for this phenomenon about LCM's Anonymous 2017-01-15 04:50:11 Post No. 8603283 [Report]

Hello guys

I have number N and want to find the LCM of all numbers from 1->20.

I figured out this methodology seems to work but cant figure out why?

First I find all the primes starting from 5 -> 20, which means this is 5,7,11,13,17.

Multiply them all together and get a lower bound for N.

Now this is the magic that I dont understand.
I take the floor of 2nd root of 20 and then floor of 3rd root of 20.

This gives me 2**4 and 3**2 multiplied by my previous lower bound.

This seems to work for me every time, I don't know if its mathematically consistent or a coincidence.

I just want to know why this works, why can a combination of 2's and 3's describe all missing factors of my number N?

Thank ypou

Anonymous 2017-01-15 04:54:06 Post No.8603287
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Anonymous 2017-01-15 04:54:06 Post No.8603287 [Report]

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>>8603283
I have one part of it figured out though:
I understand why 2**4 and 3**2 work, because for all numbers under 20, finding the nth root will give me the highest possible multiple for 2 and 3, which is 18 and 16..

What remains is I dont understand how it magically happens to find the exact correct amount of factors needed between 1 and 20.. for example 12, in which 2*2*3 was needed.

Anonymous 2017-01-15 04:58:46 Post No.8603290
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Anonymous 2017-01-15 04:58:46 Post No.8603290 [Report]

>>8603287
coincidence

Anonymous 2017-01-15 05:00:12 Post No.8603293
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Anonymous 2017-01-15 05:00:12 Post No.8603293 [Report]

>>8603290
Can you show me counter example please?

Anonymous 2017-01-15 05:52:02 Post No.8603379
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Anonymous 2017-01-15 05:52:02 Post No.8603379 [Report]

In all cases, multiplying each prime with the highest multiplicity will give you the lcm. It comes from the fundamental theorem of algebra that every number is a prime or product of primes. And the fact that the lcm is divided by each of the numbers.

Anonymous 2017-01-15 07:35:02 Post No.8603551
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Anonymous 2017-01-15 07:35:02 Post No.8603551 [Report]

>>8603379
Ok but what about this case,

Anonymous 2017-01-15 10:15:39 Post No.8603785
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Anonymous 2017-01-15 10:15:39 Post No.8603785 [Report]

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bomp

Anonymous 2017-01-15 11:00:14 Post No.8605301
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Anonymous 2017-01-15 11:00:14 Post No.8605301 [Report]

>>8603551
Write out all numbers in terms of thier primes and multiply by the highest power of each prime.

Ex. 1,2,3,2^2,5,2*3,7,2^3,3^2,5*2

Lcm = 2^3*5*7*3^2

Anonymous 2017-01-16 05:38:27 Post No.8606236
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Anonymous 2017-01-16 05:38:27 Post No.8606236 [Report]

>>8605301
Yes, but why taking the root of 21 and 3rd root of 21 give me the number of 2's and 3's required to produce the LCM.

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