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Divisibility Challenge

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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,690
Given unequal integers $a, b, c$ prove that $(a-b)^5+(b-c)^5+(c-a)^5$ is divisible by $5(a-b)(b-c)(c-a)$.
 

kaliprasad

Well-known member
Mar 31, 2013
1,309
Let us put $f(x) = (x-b)^5 + (b-c)^5 + (c-x)^5$
Putting x = b we get f(b) = 0
so (x-b) is a factor
so a-b is a factor of $(a-b)^5 + (b-c)^5 + (c-a)^5$

similarly we have (b-c) and (c-a) are factors

now $(a-b)^5 = a^5 – 5a b^4 + 10a^2b^3 – 10 a^3 b^2 +5 a^4 b – b^5 = a^5 – b^5 + 5m$ where $m = - a b^4 + 2a^2b^3 – 2 a^3 b^2 + a^4 b b^4$
similarly $(b-c)^5 = b^5 – c^5 + 5n$
$(c-a)^5 = c^5 – a^5 + 5k$
Adding we get $(a-b)^5 + (b-c)^5 + (c-a)^5 = 5 (m+n+k)$

So $(a-b)^5 + (b-c)^5 + (c-a)^5$ is divisible by $5(a-b)(b-c)(c-a)$
 
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anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,690
Let us put $f(x) = (x-b)^5 + (b-c)^5 + (c-x)^5$
Putting x = b we get f(b) = 0
so (x-b) is a factor
so a-b is a factor of $(a-b)^5 + (b-c)^5 + (c-a)^5$

similarly we have (b-c) and (c-a) are factors

now $(a-b)^5 = a^5 – 5a b^4 + 10a^2b^3 – 10 a^3 b^2 +5 a^4 b – b^5 = a^5 – b^5 + 5m$ where $m = - a b^4 + 2a^2b^3 – 2 a^3 b^2 + a^4 b b^4$
similarly $(b-c)^5 = b^5 – c^5 + 5n$
$(c-a)^5 = c^5 – a^5 + 5k$
Adding we get $(a-b)^5 + (b-c)^5 + (c-a)^5 = 5 (m+n+k)$

So $(a-b)^5 + (b-c)^5 + (c-a)^5$ is divisible by $5(a-b)(b-c)(c-a)$
Sorry for the late reply, kaliprasad and thanks for participating!:eek:

I like your solution because of the way you introduced a polynomial function for the problem and well done!(Sun)
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
Given unequal integers $a, b, c$ prove that $(a-b)^5+(b-c)^5+(c-a)^5$ is divisible by $5(a-b)(b-c)(c-a)$.
Let $S_5=(a-b)^5+(b-c)^5+(c-a)^5$, $S^3=(a-b)^3+(b-c)^3+(c-a)^3$ and $S_2=(a-b)^2+(b-c)^2+(c-a)^2$.

From this thread http://mathhelpboards.com/challenge...^5-5=-^3-b^3-c^3-3-*-^2-b^2-c^2-2-a-8276.html we know that

$$\frac{S_5}{5}=\frac{S_3}{3}\frac{S_2}{2}$$.

Note that since $(a-b)+(b-c)+(c-a)=0$, we have $S_3=3(a-b)(b-c)(c-a)$.

Thus $$S_5=5(a-b)(b-c)(c-a)\frac{S_2}{2}$$.

Clearly $2$ divides $S_2$ and thus we are done.
 
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anemone

MHB POTW Director
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Feb 14, 2012
3,690
Let $S_5=(a-b)^5+(b-c)^5+(c-a)^5$, $S^3=(a-b)^3+(b-c)^3+(c-a)^3$ and $S_2=(a-b)^2+(b-c)^2+(c-a)^2$.

From this thread http://mathhelpboards.com/challenge...^5-5=-^3-b^3-c^3-3-*-^2-b^2-c^2-2-a-8276.html we know that

$$\frac{S_5}{5}=\frac{S_3}{3}\frac{S_2}{2}$$.

Note that since $(a-b)+(b-c)+(c-a)=0$, we have $S_3=3(a-b)(b-c)(c-a)$.

Thus $$S_5=5(a-b)(b-c)(c-a)\frac{S_2}{2}$$.

Clearly $2$ divides $S_2$ and thus we are done.
Hey caffeinemachine,

Thanks for participating and that's another trick for me to learn today!(Sun)