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# Number Theory Problem

It first appeared in RMO 2012

Note by Kishlaya Jaiswal
4 years, 1 month ago

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A better solution would be choose any prime $p$ dividing $a,b,c$ and consider the maximum powers of $p$ in $a,b,c$ respectively $\alpha,\beta,\gamma$ and WLOG let $\alpha\ge \beta \ge \gamma$ .Now the max power of $p$ dividing $(a+b+c)^{31}$ is $p^{31\gamma}$ .It suffices to show that $\alpha+\beta+\gamma\le 31\gamma$ but $\alpha\le 5\beta\le 25\gamma$ from the given condition.So summing them we get the desired inequality.This helps an easy generalization: let $a,b,c$ be natural numbers such that $a\mid b^n,b\mid c^n,c\mid a^n$ then $abc\mid (a+b+c)^{n^2+n+1}$

- 4 years, 1 month ago

Well, first of all we note that a divides $$c^{25}$$: if b divides $$c^5$$, then $$b^5$$ divides $$c^{25}$$. Because a divides $$b^5$$, it divides also $$c^{25}$$. This can be applied cyclically: b divides $$a^{25}$$ and c divides $$b^{25}$$.

In the factorization of $$(a+b+c+)^{31}$$ there will be three members $$a^{31};b^{31};c^{31}$$, members like $$n\cdot a^k \cdot b^{31-k}$$ (and cyclical, i.e. others with ac or bc instead of ab) and members like $$abc \cdot something$$. Clearly, $$abc$$ divides all the members of the third type. We can express $$a^{31}$$ as $$a^{25}\cdot a^5 \cdot a$$, for the observation written above, it's dividible by $$abc$$.

Remains the second case:let's assume WLOG that's with the letters a and b.

We have three cases: k<5,k=5 and k>5.

In the first one, $$31-k\geq 26$$, so we can express $$a^k\cdot b^{31-k}$$ as $$a^k cdot b^{25}\cdot b^h$$ where h is a positive integer. a divides $$a^k$$, b divides $$b^h$$ and c divides $$b^25$$ so abc divides $$a^k\cdot b^{31-k}$$ .

In the second case,$$a^k\cdot b^{31-k}$$ is $$a^5\cdot b^{21}\cdot b^5$$. c divides $$a^5$$, b divides $$b^{21}$$ and a divides $$b^5$$.

In the third case, we can express $$a^k\cdot b^{31-k}$$ as $$a^5 cdot a^j \cdot b^{31-k}$$ where j is a positive integer. c divides $$a^5$$, b divides $$b^{31-k}$$ and a divides $$a^j$$. Q.E.D.

- 4 years, 1 month ago