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Intern  Joined: 22 Jun 2010
Posts: 36
How many zeros does 100! end with?  [#permalink]

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70 00:00

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Question Stats: 84% (00:30) correct 16% (01:08) wrong based on 1878 sessions

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How many zeros does 100! end with?

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

Spoiler: :: Solution
Find how many times the factor 5 is contained in 100!. That is, we have to find the largest such that 100! is divisible by . There are 20 multiples of 5 in the first hundred but 25, 50, 75, and 100 have to be counted twice because they are divisible by $$25 = 5^2$$ . So, the answer is 24.

I have absolutely no Idea what they are telling me...
Can someone please post a simple explanation for the rationale behind the explanation, or (even better) provide an alternative simple approach?

Thanks!

Originally posted by AndreG on 07 Sep 2010, 13:41.
Last edited by Bunuel on 20 Feb 2019, 03:42, edited 1 time in total.
Renamed the topic and edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 59587
Re: How many zeros does 100! end with?  [#permalink]

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34
31
AndreG wrote:
How many zeros does 100! end with?
• 20
• 24
• 25
• 30
• 32

expl.
Find how many times the factor 5 is contained in 100!. That is, we have to find the largest such that 100! is divisible by . There are 20 multiples of 5 in the first hundred but 25, 50, 75, and 100 have to be counted twice because they are divisible by $$25 = 5^2$$ . So, the answer is 24.

I have absolutely no Idea what they are telling me...
Can someone please post a simple explanation for the rationale behind the explanation, or (even better) provide an alternative simple approach?

Thanks!

Trailing zeros:
Trailing zeros are a sequence of 0's in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow.

Fro example, 125000 has 3 trailing zeros (125000);

The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer n, 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

It's more simple if you look at an example:

How many zeros are in the end (after which no other digits follow) of 32!?
$$\frac{32}{5}+\frac{32}{5^2}=6+1=7$$ (denominator must be less than 32, $$5^2=25$$ is less)

So there are 7 zeros in 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, this is equivalent to the number of factors 10, each of which gives one more trailing zero.

BACK TO THE ORIGINAL QUESTION:

According to above 100! has $$\frac{100}{5}+\frac{100}{25}=20+4=24$$ trailing zeros.

For more on this issues check Factorials and Number Theory links in my signature.

Hope it helps.
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Posts: 717
Location: London
Re: How many zeros does 100! end with?  [#permalink]

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feruz77 wrote:
the product of all integers from 1 to 100 will have the following number of zeros at the end:
a) 20
b) 24
c) 19
d) 22
e) 28

pls, help with solution method!

Search through the forums (read the math book). There is several threads discussing this.

The number of trailing zeros in 100! is (100/5)+(100/25)=24

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Intern  Joined: 24 Jun 2010
Posts: 11
Re: How many zeros does 100! end with?  [#permalink]

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the question can be re-framed as (100!/10^x) now find x?

100!/(2*5)^x---now factorize 100! by 2 and 5

when factorized by 5 will give the least power 24
Ans 24
Director  Joined: 23 Apr 2010
Posts: 503
Re: How many zeros does 100! end with?  [#permalink]

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Is this a relevant GMAT question? Thank you.
Math Expert V
Joined: 02 Sep 2009
Posts: 59587
Re: How many zeros does 100! end with?  [#permalink]

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nonameee wrote:
Is this a relevant GMAT question? Thank you.

Why do you think that it's not? I've seen several GMAT questions testing the trailing zeros concept (everything-about-factorials-on-the-gmat-85592.html):
factorial-one-106060.html
product-of-sequence-101187.html
how-many-zeroes-at-the-end-of-101752.html
least-value-of-n-m09q33-76716.html
number-properties-from-gmatprep-84770.html
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Re: How many zeros does 100! end with?  [#permalink]

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Hi Bunuel,

just a question that I have in mind:

You ALWAYS divide by 5? You never divide by another prime factor for those type of questions?

if yes what are the exceptions?

Thanks!
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GMAT 1: 780 Q51 V46 Re: How many zeros does 100! end with?  [#permalink]

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2
1
1
AndreG wrote:
How many zeros does 100! end with?

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

expl.
Find how many times the factor 5 is contained in 100!. That is, we have to find the largest such that 100! is divisible by . There are 20 multiples of 5 in the first hundred but 25, 50, 75, and 100 have to be counted twice because they are divisible by $$25 = 5^2$$ . So, the answer is 24.

I have absolutely no Idea what they are telling me...
Can someone please post a simple explanation for the rationale behind the explanation, or (even better) provide an alternative simple approach?

Thanks!

100! = 1 x 2 x 3 x ..... x 100

10 = 5 x 2

2s are in abundance however there is limited supply of 5s

How many multiples of 5 are there from 1 to 100- One way is counting other way is 100/5 = 20
How many multiples of 25 are there which contain an extra five = 100/25 = 4
There is no point going forward as the next power of 5 is 125 which is greater than 100.
That does it: 20 + 4 = 24
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Math Expert V
Joined: 02 Sep 2009
Posts: 59587
Re: How many zeros does 100! end with?  [#permalink]

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1
2
Paris75 wrote:
Hi Bunuel,

just a question that I have in mind:

You ALWAYS divide by 5? You never divide by another prime factor for those type of questions?

if yes what are the exceptions?

Thanks!

For trailing zeros we need only the power of 5. Check here for theory: everything-about-factorials-on-the-gmat-85592.html

Similar questions to practice:
if-n-is-the-greatest-positive-integer-for-which-2n-is-a-fact-144694.html
what-is-the-largest-power-of-3-contained-in-103525.html
if-n-is-the-product-of-all-positive-integers-less-than-103218.html
if-n-is-the-product-of-integers-from-1-to-20-inclusive-106289.html
if-n-is-the-product-of-all-multiples-of-3-between-1-and-101187.html
if-p-is-the-product-of-integers-from-1-to-30-inclusive-137721.html
what-is-the-greatest-value-of-m-such-that-4-m-is-a-factor-of-105746.html
if-6-y-is-a-factor-of-10-2-what-is-the-greatest-possible-129353.html
if-m-is-the-product-of-all-integers-from-1-to-40-inclusive-108971.html
if-p-is-a-natural-number-and-p-ends-with-y-trailing-zeros-108251.html
if-73-has-16-zeroes-at-the-end-how-many-zeroes-will-147353.html
find-the-number-of-trailing-zeros-in-the-expansion-of-108249.html
how-many-zeros-are-the-end-of-142479.html
how-many-zeros-does-100-end-with-100599.html
find-the-number-of-trailing-zeros-in-the-product-of-108248.html
if-60-is-written-out-as-an-integer-with-how-many-consecuti-97597.html
if-n-is-a-positive-integer-and-10-n-is-a-factor-of-m-what-153375.html
if-d-is-a-positive-integer-and-f-is-the-product-of-the-first-126692.html

Hope it helps.
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Re: How many zeros does 100! end with?  [#permalink]

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3
What does this question test?

It asks for the number of zeros 100! ends with. If a number ends with zero it has to be a multiple of 10. So essentially we have to find out how many times in 100! does multiplication by 10 happen. But multiplication by 10 also happens when multiplication by both 5 and 2 happen. So it is better to see the factors of the number and then find the number of times multiplication by all those factors happen.

Consider the simpler example. 4*5*6. How many zeros does it end with?

We see 2 occurs twice in 4 which is 2*2, and once in 6 which is 2*3 and 5 occurs only once. So 5 is a limiting factor. Since 5 occurs only once, the number of times multiplication by both 5 and 2 happen or in other words the number of times multiplication by 10 happens or the number of zeros the number ends with is only 1.

In the case of 100!, 5 occurs in 5, 10, 15, 20 and so on up to 100 i.e, 20 times. But remember 5 occurs twice in 25 which is 5*5 , twice in 50 which is 5*5*2 and similarly twice each in 75 and 100. So it actually occurs 24 times.

2 occurs a lot more times and so 5 is the limiting factor.

Applying the same logic as in the simpler example, the number of zeros 100! ends with is 24.
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Posts: 4
Re: How many zeros does 100! end with?  [#permalink]

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Hi Guys,

Am still abit confused about the whole concept. Some clarification would be appreciated.

With the formula, I see that the trailing zeros for the following are:
1) For 32!, (32/5) + (32/25) = 7
2) For 25!, (25/5) + (25/25) = 6
3) For 10!, (10/5) = 2
4) For 5!, (5/5) = 1

However, as i plug the numbers in and count on the calculator, the answer is different
1) For 32!, 263,130,836,933,693,000,000,000,000,000,000,000. A total of 21 trailing zeros
2) For 25!, 15,511,210,043,331,000,000,000,000. A total of 12 trailing zeros
3) For 10!, 3,628,800. A total of 2 trailing zeros
4) For 5!, 120. A total of 1 trailing zeros

Why is it that the first 2 doesnt comply whereas the last 2 does. I mean I understand the answer is correct here but am just trying to get my head around this.
Math Expert V
Joined: 02 Sep 2009
Posts: 59587
Re: How many zeros does 100! end with?  [#permalink]

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scottleey wrote:
Hi Guys,

Am still abit confused about the whole concept. Some clarification would be appreciated.

With the formula, I see that the trailing zeros for the following are:
1) For 32!, (32/5) + (32/25) = 7
2) For 25!, (25/5) + (25/25) = 6
3) For 10!, (10/5) = 2
4) For 5!, (5/5) = 1

However, as i plug the numbers in and count on the calculator, the answer is different
1) For 32!, 263,130,836,933,693,000,000,000,000,000,000,000. A total of 21 trailing zeros
2) For 25!, 15,511,210,043,331,000,000,000,000. A total of 12 trailing zeros
3) For 10!, 3,628,800. A total of 2 trailing zeros
4) For 5!, 120. A total of 1 trailing zeros

Why is it that the first 2 doesnt comply whereas the last 2 does. I mean I understand the answer is correct here but am just trying to get my head around this.

Use better calculator: http://www.wolframalpha.com
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GMAT 1: 760 Q50 V44 Re: How many zeros does 100! end with?  [#permalink]

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scottleey wrote:
Hi Guys,

Am still abit confused about the whole concept. Some clarification would be appreciated.

With the formula, I see that the trailing zeros for the following are:
1) For 32!, (32/5) + (32/25) = 7
2) For 25!, (25/5) + (25/25) = 6
3) For 10!, (10/5) = 2
4) For 5!, (5/5) = 1

However, as i plug the numbers in and count on the calculator, the answer is different
1) For 32!, 263,130,836,933,693,000,000,000,000,000,000,000. A total of 21 trailing zeros
2) For 25!, 15,511,210,043,331,000,000,000,000. A total of 12 trailing zeros
3) For 10!, 3,628,800. A total of 2 trailing zeros
4) For 5!, 120. A total of 1 trailing zeros

Why is it that the first 2 doesnt comply whereas the last 2 does. I mean I understand the answer is correct here but am just trying to get my head around this.

I think there is some problem with the calculator that you are using

32! = 263130836933693530167218012160000000
25! = 15511210043330985984000000
Both of these values comply with our understanding.

You can try the calculation here: http://www.calculatorsoup.com/calculato ... orials.php
Intern  B
Joined: 13 Feb 2017
Posts: 1
Re: How many zeros does 100! end with?  [#permalink]

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Bunuel wrote:
AndreG wrote:
How many zeros does 100! end with?
• 20
• 24
• 25
• 30
• 32

expl.
Find how many times the factor 5 is contained in 100!. That is, we have to find the largest such that 100! is divisible by . There are 20 multiples of 5 in the first hundred but 25, 50, 75, and 100 have to be counted twice because they are divisible by $$25 = 5^2$$ . So, the answer is 24.

I have absolutely no Idea what they are telling me...
Can someone please post a simple explanation for the rationale behind the explanation, or (even better) provide an alternative simple approach?

Thanks!

Trailing zeros:
Trailing zeros are a sequence of 0's in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow.

Fro example, 125000 has 3 trailing zeros (125000);

The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer n, 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

It's more simple if you look at an example:

How many zeros are in the end (after which no other digits follow) of 32!?
$$\frac{32}{5}+\frac{32}{5^2}=6+1=7$$ (denominator must be less than 32, $$5^2=25$$ is less)

So there are 7 zeros in 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, this is equivalent to the number of factors 10, each of which gives one more trailing zero.

BACK TO THE ORIGINAL QUESTION:

According to above 100! has $$\frac{100}{5}+\frac{100}{25}=20+4=24$$ trailing zeros.

For more on this issues check Factorials and Number Theory links in my signature.

Hope it helps.

Hi All,

I have found a faster approach to solve these kind of questions:

For trailing zero's: we need to check how many 5's are there in the number. So we can just divide the n by 5 and add the quotients:

100!
5 | 100
5 | 20
5 | 4
5 | 0

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Re: How many zeros does 100! end with?  [#permalink]

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AndreG wrote:
How many zeros does 100! end with?

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

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!, so the question is: how many factors 5 are there?

We can use the following trick: Divide 100 by 5 and its subsequent quotients by 5 as long as the quotient is nonzero (and each time ignore any nonzero remainder). The final step is add up all these nonzero quotients and that will be 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. We see that 20 + 4 = 24, so there are 24 factors 5 (and hence 10) in 100!. So 100! ends with 24 zeros.

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Re: How many zeros does 100! end with?  [#permalink]

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AndreG wrote:
How many zeros does 100! end with?

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

expl.
Find how many times the factor 5 is contained in 100!. That is, we have to find the largest such that 100! is divisible by . There are 20 multiples of 5 in the first hundred but 25, 50, 75, and 100 have to be counted twice because they are divisible by $$25 = 5^2$$ . So, the answer is 24.

I have absolutely no Idea what they are telling me...
Can someone please post a simple explanation for the rationale behind the explanation, or (even better) provide an alternative simple approach?

Thanks!

I took 100! in excel, and I've got this = 9.3326E+157
So, 100! ends with a lot more 0 than 24.

Did i miss something?
Really appreciated for explanation.
Math Expert V
Joined: 02 Sep 2009
Posts: 59587
Re: How many zeros does 100! end with?  [#permalink]

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2
olgaromazan wrote:
AndreG wrote:
How many zeros does 100! end with?

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

expl.
Find how many times the factor 5 is contained in 100!. That is, we have to find the largest such that 100! is divisible by . There are 20 multiples of 5 in the first hundred but 25, 50, 75, and 100 have to be counted twice because they are divisible by $$25 = 5^2$$ . So, the answer is 24.

I have absolutely no Idea what they are telling me...
Can someone please post a simple explanation for the rationale behind the explanation, or (even better) provide an alternative simple approach?

Thanks!

I took 100! in excel, and I've got this = 9.3326E+157
So, 100! ends with a lot more 0 than 24.

Did i miss something?
Really appreciated for explanation.

In short - don't trust Excel, trust experts above.

100! = 93326215443944152681699238856266700490715968264381621468592963895
217599993229915608941463976156518286253697920827223758251185210916864
000000000000000000000000
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Joined: 03 Apr 2018
Posts: 38
Re: How many zeros does 100! end with?  [#permalink]

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Bunuel wrote:
AndreG wrote:
How many zeros does 100! end with?
• 20
• 24
• 25
• 30
• 32

expl.
Find how many times the factor 5 is contained in 100!. That is, we have to find the largest such that 100! is divisible by . There are 20 multiples of 5 in the first hundred but 25, 50, 75, and 100 have to be counted twice because they are divisible by $$25 = 5^2$$ . So, the answer is 24.

I have absolutely no Idea what they are telling me...
Can someone please post a simple explanation for the rationale behind the explanation, or (even better) provide an alternative simple approach?

Thanks!

Trailing zeros:
Trailing zeros are a sequence of 0's in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow.

Fro example, 125000 has 3 trailing zeros (125000);

The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer n, 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

It's more simple if you look at an example:

How many zeros are in the end (after which no other digits follow) of 32!?
$$\frac{32}{5}+\frac{32}{5^2}=6+1=7$$ (denominator must be less than 32, $$5^2=25$$ is less)

So there are 7 zeros in 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, this is equivalent to the number of factors 10, each of which gives one more trailing zero.

BACK TO THE ORIGINAL QUESTION:

According to above 100! has $$\frac{100}{5}+\frac{100}{25}=20+4=24$$ trailing zeros.

For more on this issues check Factorials and Number Theory links in my signature.

Hope it helps.

Hi Bunuel, can you please explain why K = 2 and not k=3 this time as 125>100. (as explained in the formula above 5^(K+1)>n Re: How many zeros does 100! end with?   [#permalink] 20 Sep 2019, 06:16
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