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Mekhanik [1.2K]
3 years ago
15

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 $mnp$.
Mathematics
1 answer:
Ksivusya [100]3 years ago
3 0
Let <span>simplify the equations a^7xy-a^6y-a^5x=a^4(b^4-1) and (a^mx-a^n)(a^py-a^2)=a^4b^4:
</span>
<span>1) a^7xy-a^6y-a^5x+a^4=a^4b^4
</span>
<span>and
</span>
<span>2) a^{m+p}xy-a^{n+p}y-a^{m+2}x+a^{n+2}=a^4b^4.
</span>
<span>Equate the coefficients:
</span><span>
</span><span>xy: m+p=7  \\ y: n+p=6 \\ x:  m+2=5 \\ 1: n+2=4.</span>
Then n=2 \\ p=4 \\ m=3 and mnp=24.
<span />
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