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# the highest prime less than 50, that divides the binomial coefficient 100C50 ? a) 37, b) 31, c)47 d)43 Grade:12

## 13 Answers

9 years ago

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).

9 years ago

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?

9 years ago

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.

9 years ago

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).

9 years ago

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).

9 years ago

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).

9 years ago

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.

9 years ago

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 .. ??

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