How Many Zeros does 100! end With? GMAT Problem Solving

Question: How many zeros does 100! end with?

Correct Answer: B

Solution and Explanation:
Approach Solution 1:
Candidates are first required to understand the concept of trailing zeros. Trailing zeros can be expanded as a continuous sequence of 0’s in the form of decimal representation. These are generally represented in the form of numbers. Followed by this, there are no other digits to be utilised in the given number. Below mentioned is an example that justifies the above mentioned statement: 125000 has 3 trailing zeros (125000);

It can be highlighted that the number of trailing zeros can be indicated as n!. n! here is the factorial of a non-negative integer n, which can be determined with this formula:

\(\frac{n}{5}+\frac{n}{5^2}+\frac{n}{5^3}+...+\frac{n}{5^k}\), where k must be chosen such that 5^(k+1)>n.

Mentioned below is an example of the above mentioned equation;
How many zeros are in the end (after which no other digits follow) of 32!?

\(\frac{32}{5}+\frac{32}{5^2}\)= 6+1=7. The denominator in this case is required to be less than 32, thus, \(5^2=25\)is yielding less value.

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

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

Now, considering the original question; we can derive the below mentioned conclusion:
100! has 100/5+100/25=20+4=24 trailing zeros.

Approach Solution 2:
The problem statement asks to find the number of zeros 100! ends with.

This question can be solved with a quicker and easier approach.

We are looking the number of 2-and-5 pairs in 100! since each pair produces a factor of 10 and thus a 0 at the end of 100!.
There are more factors of 2 than 5 in 100!. Therefore, the question can be reframed as: how many factors 5 are there?

We can divide 100 by 5 and its subsequent quotients by 5 as long as the quotient is nonzero (and each time ignore any nonzero remainder).
In the final step, we need to add up all these nonzero quotients. This will give us the number of factors of 5 in 100!.

100/5 = 20
20/5 = 4
Since 4/5 has a zero quotient, we can stop here.

Therefore, by Adding the quotients we get:
20+4+0 = 24.

So there are 24 factors 5 (and hence 10) in 100!.
Therefore,100! ends with 24 zeros.

Hence, we can infer that 100! consists of 24 trailing zeros.

Approach Solution 3:
The problem statement asks to find the number of zeros 100! ends with.

100! = 1 x 2 x 3 x ..... x 100
10 = 5 x 2
2s are in abundance however there is a limited supply of 5s

To find the number of 0's in 100! let us find the number of [5's] in 100!

The number of multiples of 5 is there from 1 to 100:
One way is counting and the other way is 100/5 = 20

The number of multiples of 25 is there which contains an extra five i.e 100/25 = 4

There is no point in going forward as the next power of 5 is 125 which is greater than 100.

Hence, the number of [5's] in 100!
=> 20 + 4 + 0
=> 24

Hence, we can infer that 100! consists of 24 trailing zeros.

“How many zeros does 100! end with”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve the GMAT Problem Solving questions, the candidates must know a basic knowledge of mathematics to calculate the sum properly. The candidates can follow GMAT Quant practice papers to explore several sorts of questions that will enable them to improve their mathematical understanding.

Suggested GMAT Problem Solving Questions

SHARE ON FACEBOOK SHARE ON TWITTER

How Many Zeros does 100! end With? GMAT Problem Solving

Fees Structure

Comments

SUBSCRIBE TO OUR NEWS LETTER