Discrete math problem!What is the highest power of 5 which can be divided by 500!

[haya]

Discrete math problem!What is the largest power of 5 that divides 500!

A question at my university from discrete math. The problem is: what is the largest power of 5 that divides 500! (zero reminder).
I would appreciate your fast help, I'll be having an exam next week. :3 Thank you!

 

[HallsofIvy]

Do you understand what is being asked here? You understand that 500! is an even number, don't you? And that all powers of 5 are odd? Are you sure that the problem isn't the other way around: "what is the largest power of 5 that divides 500!?

 

[pka]

A question at my university from discrete math. The problem is: What is the highest power of 5 which can be divided by 500! (zero reminder) ---> So the problem is: 5 ^x = 0 mod(500!)
I would appreciate your fast help, I'll be having an exam next week. :3 Thank you!

Can you show a single power of five that is divisible by (5!) ?
Do you want to review the wording of the posting?
Have you inverted the order as HallsofIvy suggests?

 

[haya]

Oh my god... You are totally right, I will edit it (soory I was really tired when I wrote the post...). Yea so the problem: what is the largest power of 5 that divides 500! So: 500!=0 mod(5 ^x )

 

[pka]

Oh my god... You are totally right, I will edit it (soory I was really tired when I wrote the post...). Yea so the problem: what is the largest power of 5 that divides 500! So: 500!=0 mod(5 ^x )

The problem is trivial then. SEE HERE.

 

[haya]

Indeed it is trivial, but I have to solve this on paper. So I am curious that how you calculate it, I have to show that to the professor. Not just the result.

 

[pka]

Indeed it is trivial, but I have to solve this on paper. So I am curious that how you calculate it, I have to show that to the professor. Not just the result.

Do you have a huge amount of time to waste just to write it all out?

Can you calculate \(\displaystyle \sum\limits_{k = 1}^5 {\left\lfloor {500*5^{ - k} } \right\rfloor } =~?\)

 

[Deleted member 4993]

Indeed it is trivial, but I have to solve this on paper. So I am curious that how you calculate it, I have to show that to the professor. Not just the result.

That is the thinking part you need to do...

From 1 to 500 - how many numbers are divisible by 5¹
From 1 to 500 - how many numbers are divisible by 5²
From 1 to 500 - how many numbers are divisible by 5 and 5²

and so on......