How many of the whole numbers from 1 to 84 have no factor greater than 1 in common with 84?

Because $84=2^2\times 3\times 7$, a number clashes with 84 exactly when it is a multiple of 2, 3 or 7. Strip the primes out one at a time: half the numbers survive the 2, leaving 42; two-thirds of those survive the 3, leaving 28; six-sevenths of those survive the 7, leaving 24. In symbols, $84\times 1/2\times 2/3\times 6/7=24$. Each distractor marks where a student stopped — 42 after handling only the 2, 28 after forgetting the 7, and $36=84\times 1/2\times 6/7$ after forgetting the 3. Note that the repeated prime $2^2$ is used only once: it is the LIST of primes that matters, not their powers.