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How many trailing Zeroes does 53! + 54! have?

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How many trailing Zeroes does 53! + 54! have? [#permalink]

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New post 30 Aug 2016, 15:29
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How many trailing Zeroes does 53! + 54! have?
(A) 12
(B) 13
(C) 14
(D) 15
(E) 16




Note -> This question is a part of Stone cold's PS and DS question collection -> https://gmatclub.com/forum/stonecold-s- ... l#p1676182
[Reveal] Spoiler: OA

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Re: How many trailing Zeroes does 53! + 54! have? [#permalink]

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53! + 54! = 53! + 54 * 53!
= 53! (1 + 54)
= 53! * 55

Number of trailing 0s in 53! = number of 5s in the expansion of 53!
= 10 + 2 = 12

There is 1 more 5 in 55.

Hence, total number of trailing 0s = 12 + 1 = 13

Answer (B).

In most cases, when we are adding multiple terms, all of which have trailing 0s, the sum will have as many trailing 0s as that in the lowest term. Example: 20 + 2300 + 34210000 = 34212320 -> one 0 because lowest term 20 has one 0.

This case has an exception because of an additional 5 in 55.

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Re: How many trailing Zeroes does 53! + 54! have? [#permalink]

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New post 31 Aug 2016, 06:09
stonecold wrote:
How many trailing Zeroes does 53! + 54! have?
(A) 12
(B) 13
(C) 14
(D) 15
(E) 16




Another excellent question for practise => how-many-trailing-zeroes-does-49-50-have-217403.html?fl=similar


53!+54! = 53! (1+54) = 53!*55

Now No. of zeros in 53!*55 = pairs of 5 and 2 in prime factorization of 53!*55

Power of 5 in 53! = [53/5]+[53/5^2] = 10+2 = 12

i.e. Power of 5 in 53!*55 = 12+1(one power of 5 in 55) = 13
Since power of 2 in 53! will be much greater than power of 5 in 53!*55 so no. of pairs of 5 and 2 = power of 5 in 53!*55 = 13

Answer: Option B
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Re: How many trailing Zeroes does 53! + 54! have? [#permalink]

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New post 04 Sep 2016, 14:11
Can someone please explain why we have another trailing 0 in 55

Will it not be 00000 * 55 and this should not add another zero to the calculation??

Thanks alot

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Re: How many trailing Zeroes does 53! + 54! have? [#permalink]

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New post 04 Sep 2016, 15:00
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Tmoni26 wrote:
Can someone please explain why we have another trailing 0 in 55

Will it not be 00000 * 55 and this should not add another zero to the calculation??

Thanks alot


We don't have another trailing 0 in 55. Rather, 55 will contribute to another 0. Here's why:

The concept is based on how may pairs of 2 and 5 can be formed. Each pair will contribute one 0. Let's look at few small scale examples:

1. (22)*5 = (2*11)*5 = 110 -> one 0 because there is one 2-5 pair.
2. (220)*5 = (2*2*5*11)*5 = 1100 -> two 0s because there are two 2-5 pairs.
3. (10)*5 = (2*5)*5 = 50 -> one 0 because there is one 2-5 pair.
4. (100)*5 = (2*2*5*5)*5 = 500 -> two 0s because there are two 2-5 pairs.
5. (10)*10 = (2*5)*2*5 = 100 -> two 0s because there are two 2-5 pair.

In the problem, we have 53!*55.
55 = 5*11 -> it has no 2.
53! has a whole bunch of 2s (49 to be exact). One of the 49 2s will pair with one 5 from 55 to add another 0.

Hope this helps.

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Re: How many trailing Zeroes does 53! + 54! have? [#permalink]

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New post 05 Sep 2016, 11:44
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stonecold wrote:
How many trailing Zeroes does 53! + 54! have?
(A) 12
(B) 13
(C) 14
(D) 15
(E) 16
Another excellent question for practise => how-many-trailing-zeroes-does-49-50-have-217403.html?fl=similar


53! + 54! = 53! ( 1 + 54 )

53! ( 55 )

53! will have 12 trailing zeroes

53/5 = 10
10/5 =2

53! ( 55 ) = 53! ( 11 * 5 )

53! ( 11 * 5 ) will have 13 trailing zeroes ( 53 will have 12 trailing zeroes + 1 trailing zero for one additional 5 in 55 )

Hence correct answer will be (B) 13


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Re: How many trailing Zeroes does 53! + 54! have? [#permalink]

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New post 10 Nov 2016, 10:48
This is what i did =>
Let p=53!+54!
TO get the number of 10's we need to get the number of fives
Reason=> a 10 is formed via a 2 and a 5.
since number of 2's are abundant in quantity => no. of 5's act as the limiting agent.
p=53![1+54]=53!*55
number of 5's in 53!=> 53/5 + 53/25 +...=> 10+2=> 12
but wait 55 has one more five.
hence total number of fives are => 13
Hence number of trailing zeroes has to be 13

So its B
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Re: How many trailing Zeroes does 53! + 54! have? [#permalink]

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New post 02 Jan 2018, 04:51
stonecold wrote:
How many trailing Zeroes does 53! + 54! have?
(A) 12
(B) 13
(C) 14
(D) 15
(E) 16




Another excellent question for practise => http://gmatclub.com/forum/how-many-trai ... fl=similar


Tricky question! Though the number of 0s in both 53! and 54! will be the same because 54 doesn't bring in any additional 5, the problem is that we don't know what kind of numbers are getting added. If we are adding the numbers such as:

...200000...
...800000...

we will end up having another trailing 0 in the sum. Hence, we cannot find the number of trailing 0s from the sum.

Therefore, the step of taking 53! common becomes mandatory to ensure that we have all in factors terms.

53! * 55

As found above, we see that 53! gives 12 5s and we get another one from 55, so in all 13 5s.

Answer (B)
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Re: How many trailing Zeroes does 53! + 54! have?   [#permalink] 02 Jan 2018, 04:51
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