How to find Number of Zeroes in a Factorial Value

How to find Number of Zeroes in a Factorial Value

Number of zeroes in a factorial value is given by,

$$\frac{{\text{n}}}{5} + \frac{{\text{n}}}{{{5^2}}} + \frac{{\text{n}}}{{{5^3}}} + \,.\,.\,.\,.\,.\,.\,.$$     So on.

Where [x], denotes the greatest integer less than or equal to x.
Actually, we are dividing the given number inside the factorial by 5, 25, 125 and so on. We use this process till we don’t get division result less than 1. See the given examples below:

Find the number of zeroes in 148!
Solution:
= $$\frac{{148}}{5} + \frac{{148}}{{{5^2}}} + \frac{{148}}{{{5^3}}}$$
= 29 + 5 + 1.

[Note that we have taken integral value only not the fractional. For example $$\frac{{148}}{5}$$ = 29.6 but we have taken 29 and so on].

Step 1: We have divide 148 by 5 and get result 29.6. We only take the integral part 29 of the division result and truncate the fractional part 0.6

Step 2: We divide 148 by 25. We get result 5.92. We take 5 only and left the fraction.

Step 3: This time we divide 148 by 125, and get a result 1.184. So, integral part is 1.

Step 4: Now, we divide 148 by 625 (5⁴) but we have division result as in fraction that is 0.236. We stop our process for further as we get division result in fraction.
Thus, the total number of zeros in 148! is (29 + 5 + 1) = 35.

You can find number of zeros in any given factorial. Say, you a get to find zeros in 3689! You can find it easily by following above given process. For your convenient, I am solving it here as well to make the process more clear.

Step 1: Divide 3689 by 5, We get division result 737.8. We take 737 only.
Step 2: Divide 3689 by 25, division result is 147.56.
Step 3: Divide 3689 by 125 (5³), result would be 29.512.
Step 4: Divide 3689 by 625 (5⁴), result is 5.90.
Step 5: Divide 3689 by 3125 (5⁵), result is 1.18.
Step 6: Divide 3689 by 15625. Definitely, we are going to get a result in fraction so we leave it here.

Now, number of zeros in factorial 3689 is (737 + 147 + 29 + 5 + 1) that is 919. Note it; we don’t include fractional part in the final answer.

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