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The equation $a^7xy-a^6y-a^5x=a^4(b^4-1)$ is equivalent to the equation $(a^mx-a^n)(a^py-a^2)=a^4b^4$ for some integers $m$, $n$, and $p$. Find $mnp$.

Respuesta :

frika
Let simplify the equations [tex]a^7xy-a^6y-a^5x=a^4(b^4-1)[/tex] and [tex](a^mx-a^n)(a^py-a^2)=a^4b^4[/tex]:

1) [tex]a^7xy-a^6y-a^5x+a^4=a^4b^4[/tex]

and

2) [tex]a^{m+p}xy-a^{n+p}y-a^{m+2}x+a^{n+2}=a^4b^4[/tex].

Equate the coefficients:

[tex] xy: m+p=7 \\ y: n+p=6 \\ x: m+2=5 \\ 1: n+2=4 [/tex].
Then [tex]n=2 \\ p=4 \\ m=3[/tex] and mnp=24.

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