The sum of two numbers $p$ and $m$ is 12, and $p^3 + m^3 = 360$. We need to find the value of $pm$.

1. Problem Description

The sum of two numbers $p$ and $m$ is 12, and $p^3 + m^3 = 360$. We need to find the value of $pm$.

2. Solution Steps

We are given that

$p + m = 12$ (1)

$p^3 + m^3 = 360$ (2)

We know that:

$p^3 + m^3 = (p + m)(p^2 - pm + m^2)$

$p^3 + m^3 = (p + m)((p + m)^2 - 3pm)$

Substitute the given values into the equation:

$360 = 12((12)^2 - 3pm)$

$360 = 12(144 - 3pm)$

Divide both sides by 12:

$30 = 144 - 3pm$

Rearrange the equation to solve for $pm$:

$3pm = 144 - 30$

$3pm = 114$

Divide by 3:

$pm = \frac{114}{3}$

$pm = 38$

3. Final Answer

The value of $pm$ is 38.