Question

what is the highest power of 60 in 500! and the answer is 124 now I want to ask why do we divide 500/5+500/5^2+500/5^3 etc
because our aim is for 500! and not for 500​

Answer 1

we need to find the highest power of 60 in 500 !, don't we ?

but the question is how do we get 60 ?

let's make prime factors of 60.

i.e., 2 × 2 × 3 × 5

you see, prime factors of 60 contain 2 , 3 and 5 but if you find power of 2 or power of 3 it will definitely greater than power of 5.

it means, power of 60 depends on power of 5, don't it ?

that's why we find out power of 5 instead of 2 or 3.

now if you apply concept,

[500/5] + [500/5²] + [500/5³] + [500/5³] + ...

[.] denotes greatest integer.

= 100 + 20 + 4 + 0

= 124

Therefore the power of 60 in 500 ! is 124

I hope you got the point. :)