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Dear Bunuel, could you explain about this statement: but since there are at least as many factors of 2, this is equivalent to counting the number of factors of 10, each of which contributes one more trailing zero. ?

I did not get it. Appreciate it!!
Bunuel
Official Solution:

How many zeros are there at the end of \(100!\) ?

A. 20
B. 24
C. 25
D. 30
E. 32


To find the number of zeros at the end of 100!, we can use this calculation: \(\frac{100}{5}+\frac{100}{5^2}=20+4=24\)

Some background on trailing zeros:

Trailing zeros refer to a sequence of 0's in a decimal representation of a number, after which no other digits are present.

For instance, 125,000 has 3 trailing zeros.

The number of trailing zeros in the factorial of a non-negative integer, \(n!\), can be determined using this formula:

\(\frac{n}{5}+\frac{n}{5^2}+\frac{n}{5^3}+...+\frac{n}{5^k}\), where \(k\) must be selected such that \( 5^k \leq n\)

Let's consider an example:

How many zeros are at the end of 32!?

\(\frac{32}{5}+\frac{32}{5^2}=6+1=7\). Observe that the last denominator (\(5^2\)) must be less than 32. Also, note that we only consider the quotient of the division, that is \(\frac{32}{5}=6\).

So, there are 7 zeros at the end of 32!.

Another example: how many trailing zeros does 125! have?

\(\frac{125}{5}+\frac{125}{5^2}+\frac{125}{5^3}=25+5+1=31\),

The formula essentially counts the number of factors of 5 in n!, but since there are at least as many factors of 2, this is equivalent to counting the number of factors of 10, each of which contributes one more trailing zero.


Answer: B
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Dear Bunuel, could you explain about this statement: but since there are at least as many factors of 2, this is equivalent to counting the number of factors of 10, each of which contributes one more trailing zero. ?

I did not get it. Appreciate it!!

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The statement means that since there are always at least as many factors of 2 as there are factors of 5 in a factorial, counting the factors of 5 is enough. Each factor of 10 (which is made up of a factor of 2 and a factor of 5) contributes one trailing zero.

For similar questions check Trailing Zeros Questions and Power of a number in a factorial questions in our Special Questions Directory.

Theory on Trailing Zeros: https://gmatclub.com/forum/everything-a ... 85592.html

Hope it helps.
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