100 candles problem

It is same as "How many numbers from 1 to 100 has odd amount of positive divisors?"

All perfect squares has odd amount of positive divisors,so the answer is 1,4,9,...,100,total of 10 candles.

Before tackling the problem lets lay down some facts,

• If a candle which is lighted off it needs to have an even amount of factors*

(explanation: because the lighting and lighting off cancel each other out)

*Factors is in regard to the candle number

• So, a candle which is lighted on needs to have an odd number of factors*

So we can list down the numbers which numbers have an odd number of factors.**

** There is a faster but similar way

Before we list off to 100, we can just list off to 20. The reason being is that we need the pattern not a hardworking trial and error machine.

<-- Pause (do it on your own before i reveal the answer)-->

OK. The odd-number-factor numbers should be 1,4,9,16 We can now get the ah-ha moment. All of the numbers are perfect squares. So, we happily write down the perfect squares from 1 to 100 and count them.

But there is an even faster way! Simply take the square root of 100 which is 10 and Voila. ( In the case of an irrational number just take the [whole number] quotient.)