Which set is closed under division?

• A. Rational numbers
• B. Nonzero rational numbers
• C. Real numbers
• D. Integers

Answer: (B)

Closure under division means that if you take any two numbers from the set and the divisor isn’t zero, their quotient stays in the same set. For nonzero rational numbers, pick any two a and b with a ≠ 0 and b ≠ 0. The quotient a/b is still a rational number, and because a ≠ 0, the result isn’t zero either, so it remains a nonzero rational. That shows this set is closed under division.

This helps explain why the integer set fails: dividing 1 by 2 gives 1/2, which isn’t an integer, so the result leaves the set. The real numbers would also behave well under division as long as you’re not dividing by zero, but the nonzero rationals give a clean, straightforward example of closure under division.