Regular math compute the sum of all positive divisors of 186
1 times 186 is equal to 186 and we can also say 2 times 93 and 3 times 62 and
lastly we can also have 6 times 31 and if we sum them up we will get 284 and we are done
competition math compute the sum of all positive integers whose positive divisors sum to 186
and this is not the same as probability earlier because for the earlier was that we were given the
number 186 and then we find the sum of the numbers divisors and we saw that it was 384 but right now
we don't even know what number we are going to have and I don't even know how many of them we are
going to have at the sum of the divisors of that number were those numbers we wanted to get 186.
so how can we make this work right and as you can see for composition map you really really
have to think hard so I'm gonna show you guys my Approach for this assuming I don't know the
shortcut to find out the sum of the devices of a number then this is what I will do I'm just going
to be Begin by using a small example and let's make some observations so perhaps I'll write this
as a little observation right here suppose I have a number 40. then I would like to know the sum
of its divisor then in that case I will just say okay the sum of the devices of the number 40 well
we know one will work and two will work so I will have to add two and then I know four will
work and then five will work and then 8 times 5 and then 10 and 20 and then lastly we have 40.
okay so that will give me the sum of all the devices but I'm not just going
to add them up like this I want to look for a much better way to do this well
if you look at 40 we can prime factor it and that will give us 2 to the third power times
5 to the first Power and the truth is all these numbers can be written in the form of 2 to some
power times 5 to some power for example for the one we can write that as 2 to the 0 times 5 to
the zero that that's equal to one of course then for the two we can write it as 2 to the 1 times
5 to 0 and let's just continue to 2 times 5 to 0 and then two to zero times five to the first
and then let's just go and add two to the third times five plus zero ten two to the first that's
five first and then two to two times five first and lastly to the third that's five
notice all the powers for two has to be in between of 0 to 3 and then because right here
is five to the first then either you get zero or one for the five all right that's pretty
good but what's even better is that we can look at this expression and the factor is check this out
right here I notice that this this actually this and that they all have factors of 5 to the zero I
know it's just one but let's do this right I'm going to factor out 5 to the zero I'll put the
at the end so we get 2 to 0 plus 2 to the first plus two to the second and then two to the Third
and we multiply that by five to the zero and for the rest in fact they all have five
to the first so let's go ahead and do the same thing so we are going to get 2 to the
0 plus 2 to the one two to D2 and then two to the third and multiply by 5 to the first
in fact you see that this and that can be factored again so let's go ahead
2 to the 0 plus 2 to the one plus two to the 2 plus 2 to the third and then this then that will
just radius 5 to the 0 plus 5 to the first very nice and then I choose 40 because um it's not a
hard number to work out right and uh it's not like easy number that doesn't give me too much insight
and this right here you see if we have the Prime 2 to the powers 3 then we start with star Prime 2
and then to the zero and we add 2 to the first and so on so on so on up to 2 to the Third same thing
for the one five to the 0 plus 5 to the first so what this tells us is that if we have a number n
let's just first ready as the product Prime and we can always do this and this is called the
fundamental theorem of arithmetic I'm just going to write down P1 for the first Prime to some power
let me use Alpha One times the second prime P2 and another Power Alpha two and songs on I don't know
how many primes that we are going to get I'm just going to say p k it has to be a finite amount and
then this right here is some power let's say Alpha k based on this then I will tell you the sum of
the divisors of the number n is going to be equal to start with P1 we raise that to a zeros power
and we add P to one same Prime to the first Power and so on so on up to P 1 to the power and then
we just multiply so I will multiply that by P 2 0 plus up to P2 Alpha two and songs on song lastly
I will still have to do p k to the 0 plus p k to the first and so on so on up to p k to the alpha k
so this right here will be the quick way to find the sum of the divisors of a number but anyway
okay here is the idea I want the sum of the divisors of a number to be 186 and the formula
tells us that it's a product of this kind of Expressions so I'm just going to prime factor 186
actually just Factor not necessary prime factor yeah so now what does this tell us
well remember let's say I'm just going to use this for example first right
this three whole for the it's like this this number is just one so it's one plus some prime
that I don't know yet and this number has to be Prime and then for the 62 I would have to do
one to some power plus you know Prime to the first Power and so on so on can we achieve this entering
a prime here and then maybe a different Prime right here and again right here maybe we have more
but that's why I put on the dot can we achieve this yes in fact I can put on two twister Prime
one plus two is a prime right here and then can we achieve one plus something that gives us 62. in
fact 61 is the prime so 61 works it works wonder and we don't need a data dot in fact that's it
all right so the box that I put on right here they have to be primed because of this thing right here
okay so what does this tell me about the number n remember this 2 right here is like the P1
and then this right here is to the first power right so that tells us that we will just have P1
to the first power so for this I will tell you the numbers that we need is an equals
2 to the first Power and then we multiply that by this is my P2 and then the power which is
61 to the first Power work this out we get 122. so this right here is achievable done
now we just have to think about the other cases can we achieve this
meaning one plus something and this thing has to be a prime no so don't even think about it
because one is now Prime so that's not possible and you might be wondering how about the first one
technically I shouldn't be looking at one because that doesn't really do anything that doesn't break
down right so for this one's trick here can you think about one plus something to the first Power
and this has to be a prime plus the same Prime to the second power and in fact I don't even
know how many powers that we will have to have so I don't even know um can you think about it
the way that I did it is you just kind of test it out real quick you can try with five then you'll
see that five wouldn't work five to the third power is 125 already 5 doesn't work and of course
that means we will have up to the third power how about to
the fourth power for the fourth power you can try with three doesn't work all right
and so on so on so if you try to up to the second power you can try up to 11 13 13 wouldn't work so
yeah in fact I do not know if we can have a fast way to look at a number and ready as the sum of
P to some power I actually don't know if I can have a fast way to do it but it's not so bad
this right here I'll just tell you it's not not achievable right yeah just test it out right I'm
not going to write down all the possibilities now how about this right here 6 times 31.
this is actually possible let me show you right here again let's do the same thing 1 plus a
prime and in fact you can enter the 5 right here already so that works out wonderfully and then 1
plus something to the first plus something to the second plus something to the third plus something
to the force is this achievable yes you can just enter two that will give us 31. you can
go ahead and try it in fact this is 2 to the zero of course so this right here 2 to the zero
so this right here will tell us that and could also B this is 5 to the first power
times just look at the last one which is 2 to the fourth power and that's five and that's
eighty all to get is eighty five times sixteen so that's 80. and lastly what do we do well if
you look at this this is one plus one person work already but like this right here is that
one plus two but this right here is that no it doesn't work so this right here is also uh no
so that being said the only two numbers that will make this work in the sense that these
are the only two numbers that have the sum of their devices being called 186
so the answer will just be I'll just write it down right here the answer
for this I'll just put this down this plus that to one zero two and then we are done