7. Given that "p"is a prime number solve for "X": 1000-8x^3=8(p)(5p+2)

## Step 1: Simplify the equation<br />### Divide both sides of the equation $1000 - 8x^3 = 8(p)(5p+2)$ by 8. This simplifies the equation to $125 - x^3 = p(5p+2)$.<br /><br />## Step 2: Expand the right side<br />### Expand the right side of the equation $125 - x^3 = p(5p+2)$ to get $125 - x^3 = 5p^2 + 2p$.<br /><br />## Step 3: Rearrange the equation<br />### Rearrange the equation $125 - x^3 = 5p^2 + 2p$ to $x^3 = 125 - 5p^2 - 2p$. We are looking for integer solutions for $x$ and $p$ where $p$ is a prime number.<br /><br />## Step 4: Test prime values for p<br />### Substitute prime numbers for $p$ into the equation $x^3 = 125 - 5p^2 - 2p$ and check if the resulting value for $x^3$ is a perfect cube.<br />### If $p=2$, $x^3 = 125 - 5(2^2) - 2(2) = 125 - 20 - 4 = 101$, which is not a perfect cube.<br />### If $p=3$, $x^3 = 125 - 5(3^2) - 2(3) = 125 - 45 - 6 = 74$, which is not a perfect cube.<br />### If $p=5$, $x^3 = 125 - 5(5^2) - 2(5) = 125 - 125 - 10 = -10$, which is not a perfect cube.<br />### If $p=7$, $x^3 = 125 - 5(7^2) - 2(7) = 125 - 245 - 14 = -134$, which is not a perfect cube.<br /><br />## Step 5: Analyze the equation for larger prime numbers<br />### For $p > 5$, the right side of the equation $x^3 = 125 - 5p^2 - 2p$ becomes increasingly negative. Since we are looking for a perfect cube for $x^3$, and we have already tested smaller prime numbers, there are no prime number solutions for $p$ that yield a perfect cube for $x^3$.