Answer: because that those who don’t know what factorial is, 100! = 100 * 99 * 98 * … * 2 * 1

Ok, let’s look at how trailing zeros are created in the very first place. A trailing zero is created when a lot of of 5 is multiplied through a multiple of 2. Now all we need to do is counting the number of 5’s and also 2’s in the multiplication.

You are watching: How many zeros in 100!

Let’s count the 5’s first. 5, 10, 15, 20, 25 and also so on making a total of 20. However there is an ext to this. Since 25, 50, 75 and also 100 have actually two 5’s in every of them (25 = 5 * 5, 50 = 2 * 5 * 5, …), you need to count them twice. This makes the grand total 24. For world who prefer to look at it from a formula suggest of view

Number that 5’s = 100/5 + 100/25 + 100/125 + … = 24 (Integer values only)

Moving top top to counting the variety of 2’s. 2, 4, 6, 8, 10 and so on. Full of 50 multiples of 2’s, 25 multiples the 4’s (count these when more), 12 multiples that 8’s (count these once more) and also so on… The grand complete comes the end to

Number of 2’s = 100/2 + 100/4 + 100/8 + 100/16 + 100/32 + 100/64 + 100/128 + … = 97 (Integer worths only)

Each pair that 2 and also 5 will reason a rolling zero. Because we have actually only 24 5’s, we deserve to only do 24 bag of 2’s and also 5’s thus the number of trailing zeros in 100 factorial is 24.

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Filed in Puzzles
post by john Sonmez
Tagged easy, mathematics, numbers, puzzle

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