Number Theory. Suppose I have 2011 numbered lights in a row, all of which are of

Question

Number Theory. Suppose I have 2011 numbered lights in a row, all of which are off initially. Then I toggle all the light switches, so they are all on. Next I toggle all the even switches,so the even numbered lights are off and the odd numbered lights are still on. Then I toggle all the lights whose numbers are multiples of 3. Then I toggle all the lights whose numbers are multiples of 4. Then I toggle all the lights whose numbers are multiples of 5; 6; 7; ... ,etc. I keep going until the last step, when I toggle the 2011th light by itself. At the end of this procedure, which lights are on? How many lights are on and how many are off?
Calculus. Give the definition for a function f(x) : R -->R to be continuous at x = a and
the definition for f(x) to be differentiable at x = a. For each nonnegative integer n, define the function
fn(x) =xn sin (1/x) if x doesn't equal to 0

          0 if x=0
Where is the function fn(x) continuous? Where is the function fn(x) dierentiable? For
each nonnegative integer n, calculate the maximum integer r such that the rth derivative
fn(r) (x) exists everywhere. For each nonnegative integer n, calculate the maximum integer r
such that the rth derivative fn(r) (x) is continuous everywhere.

Explanation / Answer

For the first part, the first 1 to 44 numbers form squares that are less than 2011. This means that there are 44 lights still on at the end of all the light toggling. Proof sketch: The lights start on. The change that turns off any light occurs when there is an odd number of toggling. The light stays on for an even number of toggles which occurs for a number when it's a square of another number. I did not have time for the second part

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