Date: Thu, 20 Jul 2000 20:53:01 EDT
Subject: Factorials

Hi I was just wondering if you could tell me how many zeros are in 100,000! (factorial.) I am a college sophomore but i believe this could be considered a highschool level question. I would appreciate your help. Thank you.

sincerely
Lauren

Let's try 25! instead. This is the general idea.

What power of 10 divides 25!?

10 = 2x5 thus every time 10 divides 25! we need a 2 and a 5 to divide 25!

How many 5 divide 25!?

5 divides every 5th number (5, 10, 15, 20, 25) and 5² divides 25 thus there are 6 5's in 25!

How many 2's?

2 divides every second number, 2² divides every 4th number, 2³ divides every 8th number and 2⁴ divides every 16th number, thus the number of 2 that appear is much larger than the number of 5's.

Since there were only 25/5 + 25/5² = 5 + 1 = 6 5s in 25! we only get 10 to divide 25! 6 times in spite of the presence of all those 2s.

For 200!, 10 divides it 200/5 + 200/5² + 200/5³ = 40 + 8 + 1 = 49 times. (Notice here that 200/5³ = 200/125 = 1.6 but we want only the integer part, 1.)

For 100,000 the argument is the same.

• 5 divides every 5th number in the sequence 1, 2, ..., 100,000
• 5² divides every 25th number in the sequence 1, 2, ..., 100,000
• 5³ divides every 125th number in the sequence 1, 2, ..., 100,000
• 5⁴ divides every 5⁴th number in the sequence 1, 2, ..., 100,000

etc.

In total 100,000/5 + 100,000/5² + 100,000/5³ + ... times. (Again in each division you want only the integer part.)