Mathematical guessing

Question

A total of 2,879 votes have been distributed among the seven candidates in a one-party primary election. It is known that none of the candidates has obtained the same number of votes as another and that if the number of votes obtained by any of the candidates is divided by the number of votes obtained by any other candidate who has obtained fewer votes, the result is always a whole number. How many votes has each candidate obtained?

I was thinking of solving it by means of congruences and the Chinese Theorem of the Rest, but I don't know very well how to solve this riddle, which has been proposed to me by an acquaintance.

Hint: order the candidates vote count as $v_1<v_2<\cdots <v_7$. Then $v_1$ divides each of the $v_i$, so $v_1$ must divide the total. Reason from there. — lulu, Nov 16 '18 at 19:16
Let $v_i=n_iv_1$. Then $2879=v_1+v_2+\cdots+v_7=v_1+n_2v_1+\cdots +n_7v_1=v_1\times (1+n_1+n_2+\cdots +n_7)$ thus $v_1$ must divide $2879$. Now factor that number. — lulu, Nov 16 '18 at 19:22
Just factor $2879$. Nobody is asking you to "factor in a number you don't know." — lulu, Nov 16 '18 at 19:28
Good! So, since $v_1$ is a factor of $2879$ it can only be $1$ or $2879$. Can you eliminate one of these? — lulu, Nov 16 '18 at 19:34
Please change your title to an informative one. Nobody who has the exact problem in mind will ever find this page based on your title. — David G. Stork, Nov 16 '18 at 19:38

Henrik supports the community · Accepted Answer · 2018-11-16T21:57:24.770

Call the number of votes the candidate that gotten the fewest votes for $a_1$.

The candidate that got the second lowest number of votes must then have gotten a multiplum of $a_1$, let's define $a_2$ so that $a_2a_1$.

We can continue and define $a_3, \ldots, a_7$ similarly.

As $a_i$ for $i\geq 2$ is the factor between the number of votes for candidate $i-1$ and candidate $i$, they can't be $1$.

If we add the number of votes we get $$ 2879 = a_1+a_2a_1+\cdots+a_7a_6a_5a_4a_3a_2a_1 = a_1(1+a_2(1+a_3(1+\cdots))) $$

So we can conclude that $a_1 | 2879$, but that number is prime, so the only factors are $1$ and $2879$. $a_1=2879$ is obviously absurd, so $a_1 = 1$.

Putting that into the equation above we get $2879=1+a_2(1+\cdots)$, or $2878=a_2(1+\cdots)$. Factoring $2878$ (and ruling out the absurd possibility) we conclude $a_2=2$.

Try to go on from there.

Addition: If anyone tries to complete the calculations, the $a_i$'s are not the number of votes, and thus don't have to be different (hint: they won't be).

Ooops... let me check again. Thanks. – David G. Stork Nov 16 '18 at 20:55 — David G. Stork, Nov 16 '18 at 20:55

Mathematical guessing

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