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CAT Question: Highest Power of 3 in 1!x2!x3!x....x99!x100!

Author: Sureshbala

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Highest Power of 3 in 1!x2!x3!x............x99!x100!

Here's the solution

The above series can be written as,

1¹⁰⁰ * 2⁹⁹ * 3⁹⁸ * 4⁹⁷ *.... * 100¹
Note that for each number 'n' the power will be 101 - n

First we find the multiples of 3...
ie, 3⁹⁸, 6⁹⁵, 9⁹², 12⁸⁹, ............ , 99²

So the no. of 3's in the above numbers is
98+95+92+................+2 = 1650...--------------------->(1)

Now, we need to add multiples of 3 contained in multiples of 9...

ie, 9⁹², 18⁸³, 27⁷⁴, .............., 99²
i.e 92+83+74+........+2 = 517 (see, the power of 9 will be 9⁹² = (3*3)⁹²... similarly for 18,27,...,99)-----------------(2)

Now, add multiples of 3 in 27...

ie, 74 + 47 + 20 = 141 --------------->(3)

Finally, for 81 we have 20--------------->(4)

Add (1) (2) (3) and (4) we have, 2328...

I hope this is lucid.

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NormGreen said – Tue, 16 Sep 2008 17:03:34 -0000 ( Flag Edit Link )

Well Done It was clear and fun to try
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Sanchit kshirsagar said – Mon, 20 Oct 2008 06:45:46 -0000 ( Flag Edit Link )

I tried similar one for 4 The way I did my cals is elaborated below. Multiples of 4 are, 4^{97, 8}93, .....96^5,1001

Adding powers I got 97+93+89….. +1 = 1225.

Now adding multiples of 4 contained in multiples of 16.

16^{85, 32}69,48^53,6437,80^21,965 Adding we get 270

64 has power 37.

Total 1532. Is this calculation correct?
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Sureshbala said – Mon, 20 Oct 2008 07:22:10 -0000 ( Flag Edit Link )

Hey the above said procedure can be applied only when you are trying to calculate the highest power a number which is prime.

In case if you want to calculate the highest power of 4, since , calculate the highest power of 2, from which you can calculate the highest power of 4.

If you want to calculate the highest power of 6, since , calculate the highest power of 2 and 3. Since it is obvious that highest power of 3 will be less than that of 2 (in any factorial), the number of 6 in any factorial will be same as the number of 3 in that factorial.

Similarly, if you want to calculate highest power 10, it will be nothing but the highest power of 5.

I hope this is clear….
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Sureshbala

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nehabhansali2000 said – Wed, 24 Dec 2008 09:07:17 -0000 ( Flag Edit Link )

Hey !! I got the above mentioned calculation but can u please clarify one thing in it…

When the series given in the question is till 99!, why is 100! included in the solution..????..
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Sureshbala said – Thu, 08 Jan 2009 07:25:48 -0000 ( Flag Edit Link )

Hi, thank you and now the problem is rectified…
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saka said – Thu, 15 Jan 2009 08:59:38 -0000 ( Flag Edit Link )

stop continously chanching your pic ..it is affecting concentration.

You have not invented these concepts so don’t be over proud.

@@ SURESH BALA
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saka

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1. Sureshbala said – Fri, 16 Jan 2009 11:50:08 -0000 ( Flag Edit Link )
  
  Dear Saka,
  
  For your kind information, this is the problem that is developed by me and everyone of my students and higher authorities of my previous company know that. Also please try to understand that I am not trying to be over proud as you mentioned.
  
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2. Leonardodvin said – Fri, 16 Jan 2009 15:51:15 -0000 ( Flag Edit Link )
  
  Dear Saka,
  
  I am a student of Suresh Sir and right now faculty in one of the corporates that trains students for CAT/GMAT. In fact Suresh Sir is the one who use to train faculty for CAT/GMAT 2 years back. What he said in the above post is true.
  
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sanju09 said – Thu, 15 Jan 2009 14:11:48 -0000 ( Flag Edit Link )

This is very loutish of you, Saka! Let us know the meaning of ‘chanching’ now, will you?
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nehabhansali2000 said – Fri, 16 Jan 2009 14:17:02 -0000 ( Flag Edit Link )

I still didnt understand why did we include 100! in the solution,,,

The answer is correct ofcourse as 100 is not a multiple of 3 but what if the question asks us to dind out the highest power of 2s or 4s from the same series..??

Do we have to consider n+1 ! everytime whenever we go for a series in which n terms are given..?

Like here the question contains the series as 1x3 and in solution we have taken it as 1x3x100!
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1. Sureshbala said – Sat, 17 Jan 2009 10:09:51 -0000 ( Flag Edit Link )
  
  Dear nehabhansali,
  
  When the series is upto n!, we have to consider only upto n! and not (n+1)!. The given question is 1x…...x99...I mean it is upto 100! and not 99!.
  
  Also the above said procedure can be applied only when you are trying to calculate the highest power a number which is prime.
  
  In case if you want to calculate the highest power of 4, since 2^2 = 4, calculate the highest power of 2, from which you can calculate the highest power of 4.
  
  If you want to calculate the highest power of 6, since 6 = 2X3 , calculate the highest power of 2 and 3. Since it is obvious that highest power of 3 will be less than that of 2 (in any factorial), the number of 6 in any factorial will be same as the number of 3 in that factorial.
  
  Similarly, if you want to calculate highest power 10, it will be nothing but the highest power of 5.
  
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Invincible Ishan said – Fri, 16 Jan 2009 14:38:41 -0000 ( Flag Edit Link )

Please explain this problem 4om scratch
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Sureshbala said – Sat, 17 Jan 2009 10:52:26 -0000 ( Flag Edit Link )

Dear neha,

Initially I have considered only upto 99! in the title of the lesson and considered upto 100! in the explanation.So I have changed the title of the lesson as 1! x 2! x …..........x 99! x 100!. I think this will avoid your confusion
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sanstrom said – Tue, 20 Jan 2009 15:39:51 -0000 ( Flag Edit Link )

cool!!!!!
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asureshwaran said – Mon, 09 Feb 2009 08:11:23 -0000 ( Flag Edit Link )

good lesson man really you rock the gre and gmat and cat maths questions like just brushing your teeth

i mean you can solve problems very fast, i think it is mainly due to your experience.

anyway keep up the good work
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divyak_cs said – Thu, 23 Apr 2009 10:27:16 -0000 ( Flag Edit Link )

hey what if the series is just from 1........................50!, do we need to follow the same procedure?
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