Question

what is the highest power of 5 contained in 200 factorial

Answer 1

If you want the highest power of 5 in 200 then it is 3.292 .

Here is the explaination.

5^3= 5*5*5=125

For rest we have to know how much left to get i.e. 200/125= 1.6

So we want know 5^x=1.6

Taking log both side

We get X*log 5= log 1.6

Therefore X= log1.6/log 5= .292 ( approx)

So, (5^1)*(5^1)*(5^1)*(5^.292)= 200

Your answer is 3.292 (5^3.292=200)

Answer 2

Answer:

The highest power of 5 in 200 is 2.

Step-by-step explanation:

To find:  Highest power of 5 contained in 200!

To find out the highest power of 5 in 200! we use the formula given below.

Consider a prime number 'a'.              

The maximum power of 'a' number x in n! =  $[\frac{n}{a}]+[\frac{n}{a^2}]+[\frac{n}{a^3}]+.........$

Here, [ ] is read as the greatest integer function.

Substituting the values of 'a' and 'n' from the question, we get

The maximum power of 'a' number x in n!

                         =  $[\frac{200}{5}]+[\frac{200}{5^2}]+[\frac{200}{5^3}]+.........$  

                          = $[\frac{200}{5}]+[\frac{200}{25}]+[\frac{200}{125}]+[\frac{200}{625}]+.........$                                                            

                           = $[40]+[8]+[1.6]+[0.32].........$

                           = 40 + 8 + 1 + 0+0.....

                           = 49.

Similarly, given any prime number and asked to find out the highest power of any n! the above process is used to find the solution.

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