Which axiom states that if each side of an equation is divided by the same number, the equation remains equal?

The axiom that states if each side of an equation is divided by the same number, the equation remains equal is the Division axiom. This principle is fundamental in algebra and ensures that the balance of an equation is preserved when both sides are manipulated in the same way. For example, if you have the equation (a = b) and you divide both sides by a non-zero number (c), the resulting equation (\frac{a}{c} = \frac{b}{c}) will still hold true. This axiom underlies many algebraic operations and is essential for solving equations and understanding their properties.

The other axioms pertain to different operations: the Addition axiom involves adding the same number to both sides, the Subtraction axiom involves subtracting the same number from both sides, and the Multiplication axiom involves multiplying both sides by the same number. Each of these axioms maintains equality, but only the Division axiom focuses specifically on the operation of division.