MY CART (5)

Use Coupon: CART20 and get 20% off on all online Study Material

ITEM
DETAILS
MRP
DISCOUNT
FINAL PRICE
Total Price: R

There are no items in this cart.
Continue Shopping
Menu
rituparna chakraborty Grade:
        


the highest prime less than 50, that divides the binomial coefficient 100C50 ? a) 37, b) 31, c)47 d)43


4 years ago

Answers : (13)

Chetan Mandayam Nayakar
312 Points
										

100! contains 47 and 94, 50!*50! contains 47 twice. Therefore, answer is 47

4 years ago
Ashwin Muralidharan IIT Madras
290 Points
										

Hi Rituparna,


 


For the greatest prime number, say p.


3p < 100


 


Because 100C50 = 100!/(50!*50!)


Each of the 50! would cancel the prime numbers in 100! twice. So there has to be one more prime for it to divide 100C50.


So 3p<100. Only possibility p=31.


 


Option B.


 


Regards,


Ashwin (IIT Madras).

4 years ago
rituparna chakraborty
14 Points
										

yeah, the anwer is 31.... ur logic is right .... actually there is one more method using Legendre theorem..... would u know it by any chance?

4 years ago
Jit Mitra
25 Points
										

I am guessing 33 should be the required answer. Because exponent of 33 in 100! is 3 and in 50!^2 is 2. But that isnt in option.

4 years ago
rituparna chakraborty
14 Points
										

yeah the answer is b), thanx, but there is one more way using Legendre .... would u know it by any chance?

4 years ago
Jit Mitra
25 Points
										

I have never heard of Legendres theorem. But is it this one by any chance (i dont know the name)


 


Exponent of p in n! = [n/p] + [n/p2] + [n/p3] + ....


 


If yes, then i have used this.

4 years ago
Ashwin Muralidharan IIT Madras
290 Points
										

Hi Rituparna,


 


Yes you can use Legendre Theorem here.


It states the highest power of a prime number "p" in N! would be = Σ[N/pk]. where k = 1,2,..... and [.] denotes greatest integer function.


For example the power of 5 in 100! would be [100/5] + [100/25] + [100/125] +.... = 20+4 = 24. ie 100! will have 5^24.


 


So in this case check the prime number, for which you get the exponent to be three or more.


Only 31 will satisy that condition.


 


Hence option (B) using Legendre Theorem !!.


 


Regards,


Ashwin (IIT Madras).

4 years ago
Ashwin Muralidharan IIT Madras
290 Points
										

Hi Jit Mitra,


 


Yes that is one of Legendre Theorem.


But it is applicable only on prime numbers.


And 33 = 3x11 is not a prime number.


 


And hence 31 is the largest prime number.


 


Regards,


Ashwin (IIT Madras).

4 years ago
Ashwin Muralidharan IIT Madras
290 Points
										

If one needs to get the exponent of 33, then one has to split it into prime numbers first and then get the exponent.


 


33=3x11.


 


exp of 3 would be [100/3] + [100/9] + [100/27] + [100/81] = 33+11+3+1 = 48.


And exp of 11 would be [100/11] = 9.


 


So exp of 33 would be the lesser of the two.


Hence exponent of 33 in 100! would be 9.


 


Regards,


Ashwin (IIT Madras).

4 years ago
Zubeen Tolani
19 Points
										

ys the ans wuld b 47.


thats bcoz, 100C50 = (51*52*53*......*100)/50!   and hence any num having a factor in series in numerator wuld divide.


also as per given terms in ques, 47 wuld be the correct 1.

4 years ago
rituparna chakraborty
14 Points
										

Thanks a lot!!

4 years ago
Jit Mitra
25 Points
										

Thanks sir. I missed that.

4 years ago
Zubeen Tolani
19 Points
										

actually i m a bit weak at these binomial cofficients nd all .


so i have a kind request to ashwin sir ..... sir can u pls give some 4-5 relevant nd good questions based on the same or a bit diffrent concept .. ??

4 years ago
Think You Can Provide A Better Answer ?
Answer & Earn Cool Goodies
  • Complete JEE Main/Advanced Course and Test Series
  • OFFERED PRICE: R 15,000
  • View Details
Get extra R 3,000 off
USE CODE: MOB20
Get extra R 440 off
USE CODE: MOB20

Ask Experts

Have any Question? Ask Experts

Post Question

 
 
Answer ‘n’ Earn
Attractive Gift
Vouchers
To Win!!! Click Here for details