To solve for $M$ in the equation $F = \frac{G \cdot m \cdot M}{d^2}$, you can rearrange the formula as follows:

$M = \frac{F \cdot d^2}{G \cdot m}$

This isolates $M$ on one side of the equation.

Steps:

1. Multiply both sides of the equation by $d^2$ to get rid of the denominator: $F \cdot d^2 = G \cdot m \cdot M$
2. Divide both sides by $G \cdot m$ to solve for $M$: $M = \frac{F \cdot d^2}{G \cdot m}$

This gives the value of $M$ in terms of $F$, $G$, $m$, and $d$.

Would you like further details or have any questions?

Here are five related questions:

1. How does changing the distance $d$ affect the value of $M$?
2. What is the significance of the gravitational constant $G$ in this equation?
3. How can this formula be applied to determine the mass of a planet?
4. If the force $F$ doubles, what happens to the mass $M$?
5. How would you solve for $m$ instead of $M$ in this equation?

Tip: When dealing with gravitational force equations, always ensure your units are consistent, especially when using the gravitational constant $G$.