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Powers of a Number Contained in a Factorial
by Total Gadha - Tuesday, 9 October 2007, 03:31 AM
  cat 2009 cat 2010 finding the powers of a number in a factorial

I shall have to end here and leave the rest of it for my CBT Club students. I shall cover some problems based on this in the CBT Club this week.

 

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Re: Powers of a Number Contained in a Factorial
by chandu velamati - Tuesday, 9 October 2007, 09:52 AM
  Nice article. One of the complicated concepts made so simple. keep up the good work TG. 
Re: Powers of a Number Contained in a Factorial
by Prabhu chander Murugesan - Monday, 15 October 2007, 08:21 PM
  Really im wondering how simple u make these concepts (even if i know those earlierwink)..... Nice article TG..... keep ur good work going......
Re: Powers of a Number Contained in a Factorial
by Anish Rai - Tuesday, 16 October 2007, 12:05 PM
  Very nice.....Thanks a lot for this
Re: Powers of a Number Contained in a Factorial
by Paritosh Debnath - Wednesday, 14 November 2007, 04:33 PM
 

I have one question about " Find the highest power of 72 in 100!" I think there is no need of finding highest power of 8 as the highest power of 9 will be less than that of 8 and so the highest power of 9 will be the hightest power of 72 in 100!.

Re: Powers of a Number Contained in a Factorial
by Amit ... - Wednesday, 14 November 2007, 06:27 PM
  Hi TG,

I was aware of these ways but reading your article gave me all the confidence I need as of now...

Regards
AM
Re: Powers of a Number Contained in a Factorial
by ambar patil - Saturday, 22 March 2008, 12:43 PM
  nice .... your articles really make things seem simple n easy . Kudos approve
Re: Powers of a Number Contained in a Factorial
by harpreet Gandhi - Thursday, 27 March 2008, 03:38 PM
 

Thankx TG..

i have been following ur articles..n i must say it has helped me a lot to understand the finer points..

Thankx a lot...

N yeah Please keep adding stuffs like this..

Re: Powers of a Number Contained in a Factorial
by VIKAS NIGAM - Tuesday, 1 April 2008, 04:44 PM
 

Hi TG,

  Thats a nice article. Could you please elaborate how can we find the number of zeroes at the end of  n! if n! is expanded in some base 'k' other than 10 (where k may be a prime number, a composite number or a perfect square or a perfect cube of some prime or composite number) ? e.g. How many zeroes will be there in the end of 25! when expanded in base 14 ?

Vikas  

Re: Powers of a Number Contained in a Factorial
by TG Team - Wednesday, 2 April 2008, 06:46 PM
 

Hi Vikassmile

I am trying to answer your query. I know that it was not asked from me but i don't mind.

How many zeroes will be there in the end of 25! when expanded in base 14 ?

Look how do we convert a number from base 10 to any other base. We divide the number by required base and whatever is the remainder is the unit digit of the number in that base and the quotient obtained is further divided to get the ten's digit and soon.

Now there will be as many trailing zeroes in the expansion of a number of base 10 in some other base as many times the original number is divisible by the given base. Or in other words the highest power of the base number that can divide the given number.

In this case our number is 25! and we want to find how many trailing zeroes will be there when this number is expanded in base 14. That means what is the highst power of 14 that can divide the number i.e. = 3smile

Re: Powers of a Number Contained in a Factorial
by Total Gadha - Thursday, 3 April 2008, 02:05 PM
  Thanks Kamal smile
Re: Powers of a Number Contained in a Factorial
by srikar 2097 - Wednesday, 16 April 2008, 04:10 PM
  This is a reply to Kamal's answer. Should it not be "1" trailing zero and not "3".

as 14 = 2 X 7

Highest powers of 2 in 25! = 22
Highest powers of 7 in 25! = 1

As n(powers of 7)
Re: Powers of a Number Contained in a Factorial
by swastik shetty - Saturday, 3 May 2008, 11:54 PM
  p*q*r*s=10!  Then what is the least possible value of p+q+r+s

Ans is 175.........
Can any one explian how to solve this
Re: Powers of a Number Contained in a Factorial
by Total full - Saturday, 10 May 2008, 10:21 PM
 

Buddy..

Highest power of 7 in 25 would be calculated as :

[25/7]+[25/49]= 3+0

which comes to be 3smile!!!

Re: Powers of a Number Contained in a Factorial
by Aditya Zutshi - Saturday, 31 May 2008, 04:45 PM
 

Swastik,

If x1*x2*x3*.....*xn=C, where C is a constant, then the least value of

x1+x2+x3+.......+xn is when x1=x2=x3=......=xn.

Applying same logic here, least value pf p+q+r+s occurs when p=q=r=s therefore p^4=10!. Solving this we get, p= 43.64

Therefore, p+q+r+s=174.58 ~= 175.

But I think in CAT, such a complicated number as 10! would not come. Calculating fourth root of 10! is not a joke... Am I right TG??? Please clarify !!! smile

Re: Powers of a Number Contained in a Factorial
by varun mishra - Friday, 20 June 2008, 03:43 PM
  its gud
Re: Powers of a Number Contained in a Factorial
by nishant mittal - Sunday, 20 July 2008, 07:39 PM
 

THANK YU SIR FOR DIS WONDERFUL COMPOSITION

CUD YU DO ONE FAVOUR FOR ALL OF US..

PLEEEZ KEEP 1 COPY CAT FREE FOR ALL OF US..

 

Re: Powers of a Number Contained in a Factorial
by Varun K R - Thursday, 24 July 2008, 01:06 PM
  I have a relatively simple question I presume, but I am not able to figure out the solution
Find max n such that 42*57*92*91*52*62*63*64*65*66*67 is perfectly divisible by 42^n
please can anyone reply as to go about solving such kinda question?
Thanks in advancesmile
Re: Powers of a Number Contained in a Factorial
by rohit dwivedi - Saturday, 26 July 2008, 03:25 PM
 

Thanks,TG

on behalf of all masses like me who cant afford to go coachings either due to time or money ,for providing such a nice stuff .

 

 

Re: Powers of a Number Contained in a Factorial
by rahul ojha - Sunday, 27 July 2008, 05:18 PM
 

good, but more examples must br given

 

Re: Powers of a Number Contained in a Factorial
by manish sharma - Tuesday, 29 July 2008, 07:08 PM
 

Hi TG,

Thanks for such a nice article.

I wonder whether we can't find the highest power of 72 in 100! using the first method. [ the way we did for highest power of 30 in 50! ] ?

A-> factors of 72 are 2*2*2*3*3. Since 3 is the highest foctor, we find the highest power of 3 in 100! = 47. So 47 should be the answer.

?? Am i getting things wrong somewhere ?

 

Re: Powers of a Number Contained in a Factorial
by Samuel kumar - Wednesday, 30 July 2008, 09:54 AM
 

hi Manish,

In the first method we find the highest power of a in x!

In the second method we find the highest power of an in x!

 

Since 72= 2 X 3 , we have to use the second method. Also there are enough 2's in 100! so we simply find the no. of 3's in 100! using the second method which will come to 24.

 

Re: Powers of a Number Contained in a Factorial
by meenakshi soni - Monday, 4 August 2008, 06:45 PM
 

Hi TG,

That's really a nice article. but i have a doubt in one of the examples given above. the one in which we find the no. of divisors of 15!. In the example, first we find the indiviual highest powers of all the prime factors in 15!. Therefore we have 15! = (2^11) * (3^6) * (5^3) * (7^2) * (11) * (13).

And the no of divisors is calculated as : 12 * 7 * 4 * 3 * 2 * 2 = 4032.

i am unable to understand this step, could u please help.

Thnx in advance.

Re: Powers of a Number Contained in a Factorial
by Rahul Khandelwal - Friday, 8 August 2008, 05:25 PM
 

Hi

If i am not wrong if

n! = X^a * Y^b * Z^c

then no. of divisors = (a+1)(b+1)(c+1)

smile

rahul

Re: Powers of a Number Contained in a Factorial
by sharon koshy - Sunday, 10 August 2008, 02:53 PM
  HEY VARUN....NE IDEA WHAT THE ANSWER IS?????????????I THINK  n=3..........
Re: Powers of a Number Contained in a Factorial
by Ashwin kumar - Wednesday, 27 August 2008, 03:08 PM
  yeah Rahul u r rite..... (a+1)(b+1)(c+1)....this includes the no. itself....if u exclude the no.  then it is...[(a+1)(b+1)(c+1)-1]
Re: Powers of a Number Contained in a Factorial
by venkat s - Wednesday, 10 September 2008, 08:30 PM
 

Hi Tg thanks for your great materials they are very useful

can u please explain this

find the highest power 0f 12 that divides 5^36-1

Re: Powers of a Number Contained in a Factorial
by Sibsankar Dasmahapatra - Friday, 12 September 2008, 12:38 PM
 

Dear Varun

                             The answer is 3....

The process is ....42=2*3*7......as u have to find the max power of 42 ; so u have to check the max power of 2,3,7...in this case it is clear the max power of 7 will be the answer (..i think that u came to know from the above theroy and example also...)...so no nessary to find the power of 2,3....now to find the max power 7 , just consider the multiple of 7 in the given number...here 42,91,63....so the answer is 3......

 

Thanks

Sibu...

                

Re: Powers of a Number Contained in a Factorial
by saitu gupta - Tuesday, 23 June 2009, 12:13 PM
  Hi TG

IF U CAN PLEASE TELL ME ABOUT HOW TO CALCULATE THE TOTAL, EVEN AND ODD NO.OF FACTORS OF THE NUMBER "N" AS WELL AS THE NO. OF TRAILINGS ZEROS AND ALSO PLEASE ABOUT THE LAST 1,2,3 NON ZERO DIGITS OF THE NUMBER "N" WHERE "N"= (136!)^136!

IT WOULD B SO NICE OF U..THNX IN ADVANCE....


WAITING 4 UR REPLY smile
Re: Powers of a Number Contained in a Factorial
by Software Engineer - Tuesday, 23 June 2009, 12:12 PM
  Saitu, I hope this will help you smile
http://totalgadha.com/mod/forum/discuss.php?d=5224
Re: Powers of a Number Contained in a Factorial
by Sudhanshu Trivedi - Monday, 7 September 2009, 11:55 AM
  Hello Aditya,
See this problem can be solved easily with the AM >= GM approach.
(p+q+r+s)/4  >= (pxqxrxs)^0.25

10! can be broken down into the various prime factors by finding their highest powers each,agreeing to your point that such complicated fourth roots of a number would not be asked in CAT,but they might throw an even bigger number with a cleaner 4th root.

Do tell me if I am wrong anywhere in my approach.
Re: Powers of a Number Contained in a Factorial
by Pallav Jain - Monday, 26 October 2009, 01:19 PM
  Hi Guys,

Can u solve this question?

What is the product of all factors of the number N = 64 x 102, which are divisible by 5?

  12210 × 3102 × 5140
  22210 × 3140 × 5105
  32140 × 3210 × 5102
  42140 × 3102 × 5210
  52102 × 3210 × 5140

Regards

Pallav
Re: Powers of a Number Contained in a Factorial
by ROHIT K - Tuesday, 27 October 2009, 08:41 PM
  Hi pallav,

is option (2) the correct answer?

Rohit
is there DI Lesson
by Pankaj Kumar - Friday, 30 October 2009, 11:27 AM
  HI TG,

we have here Quant Lessons and Verbal Lessons, two dedicated forum with wonderful techniques for these two sections on various topics, but I can't see sth like DI Lessons. or its thr?, if not can we expect the same as it seems missing part here.

Thanks,
Re: is there DI Lesson
by karan khetan - Friday, 6 November 2009, 09:44 PM
  u have to select even numbers from 1 to 25 and odd numbers from 26 to 200 and if u take a product of them how many zeros will be there in the product...can someone answer this question...???
Re: is there DI Lesson
by ROHIT K - Friday, 6 November 2009, 11:59 PM
  hi karan,
plz chk d previous thread where you posted this question.

Rohit
Re: Powers of a Number Contained in a Factorial
by Ashish Sharma - Saturday, 7 November 2009, 03:26 PM
  Hello pallav,

1) factors with 5^2 will be 35( (6+1)(4+1)). so total contribution to the power of 5 in the product =70
2) factors with 5 will also be 35. so total power to the power of 5 in the product=35.

power of 5 =35+70=105. only option 2 satisfies this
Re: Powers of a Number Contained in a Factorial
by amar goswami - Sunday, 15 November 2009, 01:19 AM
  Hi Pallav,

  Answer is  2210 × 3140 × 5105.

Regards
Amar
Re: Powers of a Number Contained in a Factorial
by manu thomas - Sunday, 21 November 2010, 12:04 AM
  Another method to find the right most digit of 15!

The last digit of 10! is 8.(should remember)
Now.10!*11*12*13*14*15
8*11*12*13*2*7*3*5(NOW remove all 2's corresponding to the number of 5's)
ie 8*11*12*13*7*3)=8*6=48 ie 8 is the last digit



Another eg.25!

last digit of 10!=8
11! to 20! =8
now
21*22*23*24*25=
21*2*11*23*2*12*5*5(remove corresponding 5's and 2's)
21*11*23*12=6
last digit is 8*8*6=4
Re: Powers of a Number Contained in a Factorial
by shashi vashisth - Sunday, 16 October 2011, 04:32 PM
  I am  still not getting it.-please help

Re: Powers of a Number Contained in a Factorial
by TG Team - Tuesday, 18 October 2011, 12:28 PM
 

Hi Shashi smile

I hope you are talking about this problem: How many zeroes will be there in the end of 25! when expanded in base 14 ?

Just take a much simpler case. If in place of base-14, it is base-10 then how do you find number of trailing zeroes - by finding highest power of 10 that divide the number completely. Isn't it?

Exactly same way when you want to findout number of trailing zeroes in base-14, then you just need to find that what is the highest power of 14 that divides the number completely.

Kamal Lohia