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# What is the greatest power of 5 that divides 80! exactly. (CAT, 2016)

- May 14, 2018
- Posted by: allexamshelps
- Category: CAT

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**Solution:**

**What is the greatest power of 5 that divides 80! exactly:**

**n! is called factorial n, or n factorial where n is a number.**

**If n=4, then 4! means the product of all the natural numbers from 1 to 4.**

**That is, 4!= 4 x 3 x 2 x 1; (0!= 1 and 1!=1)**

**To solve such question what we do is that we find the number of powers of the base number in n factorial (n!):**

**For example:**

**What is the greatest power of 5 that divides 25! Exactly**

**Here, we have to find the number of powers of 5 in to 25.**

** \frac { 25 }{ 5 }=5, and \frac { 25}{ { 5 }^{ 2} }=1**

**So in the first attempt, we have 5 powers and in the second when 5**^{2} was the denominator, we found 1 power.

^{2}was the denominator, we found 1 power.

**There are a total of 5+1= 6 powers of 5 in 25!.**

**Let’s check it.**

**\frac { 25! }{ { 5 }^{ 6 } } =\frac { 15511210043330985984000000 }{ 15625 } = 992717442773183102976**

**Hence, 25! is completely divisible by 5**^{6}.

^{6}.

**In the same way:**

**In 80!**

** \frac { 80 }{ 5 }= 16 and \frac { 80 }{ { 5 }^{ 2} }=3 (only integer values, quotients, are taken)**

**5**^{19} will divide 80! exactly

^{19}will divide 80! exactly

**Hence the option (C) is correct.**

** (Please leave your comments, to encourage us, to let us know how good/bad we are doing it (Please support your comments with reasons)).**

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