Introduction:

The statement to be evaluated is whether there exist distinct whole numbers a, b, and c such that a ÷ (b ÷ c) = (a ÷ b) ÷ c.

Evaluation:

To evaluate this statement, we will start by simplifying both sides of the equation:

a ÷ (b ÷ c) = (a ÷ b) ÷ c

Multiplying both sides by c, we get:

a ÷ (b ÷ c) x c = (a ÷ b) ÷ c x c

Simplifying the left side:

a ÷ (b ÷ c) x c = a ÷ (b ÷ c) x (c ÷ 1)

Using the division of fractions rule, we can simplify the left side further:

a ÷ (b ÷ c) x c = a ÷ (b ÷ c) x (c ÷ 1)

a ÷ (b ÷ c) x c = a x c ÷ b

Now simplifying the right side:

(a ÷ b) ÷ c x c = a ÷ b x (c ÷ 1)

(a ÷ b) ÷ c x c = a x (c ÷ b)

Therefore, the equation becomes:

a x c ÷ b = a x (c ÷ b)

Multiplying both sides by b gives:

a x c = a x (c ÷ b) x b

a x c = a x c

This implies that a = 0 or c = b. However, since a, b, and c are assumed to be distinct whole numbers, a cannot be 0, and therefore, c must equal b.

Conclusion:

Therefore, we can conclude that there do not exist distinct whole numbers a, b, and c such that a ÷ (b ÷ c) = (a ÷ b) ÷ c.