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Factorial

The factorial 5! means the product of all natural numbers from 5 in reducing order up to 1.
Hence 5!=5×4×3×2 1=120.
In general, n! = n × (n – 1) × (n – 2) × (n – 3) × ... 1.

Also, 0! = 1.

Sums on factorials are asked in exams.
An illustration is given below to explain how these sums are done.

 

Illustration 25

If 146! is divided by 6n, then what is the value of n?

Solution

The question here is how many 6 are there in 146!
Since 6 has 2 factors, 3 and 2, we can find the number of 6 used if we can find the number of 3 in 146!
This is given by: 146/3+146/32 +146/33 +146/34 =48+16+5+1=70.
Note that we divide with powers of the denominator as long as the numerator can be divided to yield a whole number.

 

Illustration 26

Find the number of zeroes in 128!

Solution

Note that a 0 can be the result only when 5 is multiplied by an even number.
Since the even numbers is not a constraint, our job is to find the number of 5 in 128!
This is given by: 128/5 + 128/52 + 128/53 = 25 + 5 + 1 = 31.





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