For nonzero a, which statement is always true?

For nonzero a, which statement is always true?

• A. a ÷ a = 1
• B. a × 0 = a
• C. a ÷ a = 0
• D. a × a = a

Answer: (A)

Dividing a nonzero number by itself always gives 1. This works because dividing by a is the same as multiplying by its reciprocal: a ÷ a = a × (1/a) = 1, since 1/a exists when a ≠ 0. The other statements don’t hold for every nonzero a. a × 0 equals 0, not a. So a ÷ a isn’t 0. Also, a × a equals a only when a^2 = a, which forces a to be 1 (or 0), but a is nonzero here, so it isn’t always true.

Dividing a nonzero number by itself always gives 1. This works because dividing by a is the same as multiplying by its reciprocal: a ÷ a = a × (1/a) = 1, since 1/a exists when a ≠ 0.

The other statements don’t hold for every nonzero a. a × 0 equals 0, not a. So a ÷ a isn’t 0. Also, a × a equals a only when a^2 = a, which forces a to be 1 (or 0), but a is nonzero here, so it isn’t always true.