$Find 10 distinct positive integers such that each of them divides the sum of these 10 integers$

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Find 10 distinct positive integers such that each of them divides the sum of these 10 integers. Moreover, explain how you found the solution in the most detailed way possible.

Combinatorics

Lennythesniper

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Daniel90

Your offer is way too low.
Alessandro Iraci

I second this. You need to understand that writing out plenty of details takes time, and time of qualified professionals is expensive. For 3$ the best you can get is just the answer, with no explanation, if the question is sufficiently easy (and this seems not to be, by the way).
Lennythesniper

Ok IDK how to delete questions but I just found the answer myself so never mind this question. First to submit any answer gets the money.
Alessandro Iraci

You can just let it expire and get a refund. I'll submit a feedback saying that a "delete question" feature might come in handy, though.
Alessandro Iraci

Also, I have to apologise, the question is indeed fairly simple now that I think about it. Not that it matters since you got there on your own, but I think 5-6$ would probably have been fair for this one. Sometimes it's difficult to estimate how hard a problem is before actually solving it.

Alessandro Iraci

I second this. You need to understand that writing out plenty of details takes time, and time of qualified professionals is expensive. For 3$ the best you can get is just the answer, with no explanation, if the question is sufficiently easy (and this seems not to be, by the way).
Lennythesniper

Ok IDK how to delete questions but I just found the answer myself so never mind this question. First to submit any answer gets the money.
Alessandro Iraci

You can just let it expire and get a refund. I'll submit a feedback saying that a "delete question" feature might come in handy, though.
Alessandro Iraci

Also, I have to apologise, the question is indeed fairly simple now that I think about it. Not that it matters since you got there on your own, but I think 5-6$ would probably have been fair for this one. Sometimes it's difficult to estimate how hard a problem is before actually solving it.

Answer 1

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I started by finding a simplier case, with just a set of 3 numbers:

$1 + 2 + 3 = 6$

Now I want to add another number $n$ to this set, but making sure that $6+n$ is still divsiable by $1, 2, 3$ and $n$:

$n = 4:\\ 1 + 2 + 3 + 4= 10 \rightarrow 3\nmid10$

$n = 5:\\ 1 + 2 + 3 + 5 = 11 \rightarrow 2,3,5\nmid11$

$n = 6:$
*$1 + 2 + 3 + 6 = 12$*

Notice that we just added the sum of our previous set and got a new set, which still satisfies our requirements. Adding 12 to our new set yields a newer set:

$1 + 2 + 3 + 6 + 12 = 24$

And again this newer set is satisfactory. I can then repeat this process until I have 10 distinct numbers:

$1+2+3+6+12+24+48+96+192+384=768$
This method can be generalized to get a set of $N$ positive integers:

$S = \{d_1, d_2, d_3,..., d_{k}, p, 2p, 4p, 8p, ..., 2^{N-k-1}p\}$

Where $d_1 + d_2 + d_3 + ... + d_k = p$ and $N > k$.
This is chiefly in the realm of "psuedoperfect numbers". A number is considered "perfect" if its equal to the sum of its divsors (not including itself). 6 is the simplest perfect number, and they exponentially increase from there (https://oeis.org/A000396). Meanwhile, a number is considered "psuedoperfect" if its equal to the sum of not of its divisors.

Lhs1002

$Find 10 distinct positive integers such that each of them divides the sum of these 10 integers$

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