Question

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 $mnp$.

Answer 1

Let 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`:

1)
`a^7xy-a^6y-a^5x+a^4=a^4b^4`

and

2)
`a^(m+p)xy-a^(n+p)y-a^(m+2)x+a^(n+2)=a^4b^4`.

Equate the coefficients:


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