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Prime factorize

72

8 * 9 

2³ * 3²

Factor out a 6²

2 * 6²

Make system of equations

2x + 2y = 2

x + y = 1

Solving, we get x = 1/2 and y = 1/2

Use Binary sequences ( a sequence of ones and zeroes) to help visualize.

Someone flips the switch on every single lightbulb

Every lightbulb starts turned on. ON is 1, OFF is 0.

1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

Someone flips the switch on every other lightbulb
Someone flips the switch on every third lightbulb
Someone flips the switch on every fourth lightbulb

This means that LCM of 2, 3, 4,. Which is 12.

A binary sequence of 12 terms would repeat in the a hundred light bulbs.

So lets have 12 light bulbs.

101010101010 Someone flips the switch on every other lightbulb

100011100011 Someone flips the switch on every third lightbulb

100111110010 Someone flips the switch on every fourth lightbulb

There are 7 out of 12 lightbulbs that are turned on.

Now we have a pattern of 100111110010 repeating until the number of terms is 100. We have to find the number of 1s in that 100-length pattern.

Every 12 terms that are seven 1s.

So 12 * 8 = 96

8 * 7 = 56

So out of 96 terms, 56 are 1s.

There are 4 more terms.

Counting up the pattern 100111110010 four terms, we have two 1s.

So 56 + 2 = 58

58 light bulbs are turned on

lol i had a solution ready, then deleted it because I thought it continued to 100. nuuuuuuuu

EDIT: I remade it,

 below is the explanation

Sorry, I tried to edit my answer, but Web2.0 didn't is glitching or something. I can't publish my answer, it just reverts back.

Just look at EP's answer, that is the correct method.

Substitute y

So now you have:

x³=x²+4x

0 = x³ - x² - 4x

This is a polynomial with a degree of 3, therefore it must have 3 solutions.

If it has 3 solutions, it has 3 intersections in the graph.

Does the flipping only go to the 4th light bulb?

Or does he keep flipping until he flips all 100 lightbulbs?

Yeah, I just set two equal, no need to overthink it.

Nice, hard work!

By the way, I like your riddles, it helps me warm up my brain and makes me feel mentally healthy lol.