The largest power of 2 that divides frac{200! | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) To find the largest power of $2$ that divides $\frac{200!}{100!}$,we use Legendre's Formula,which states that the exponent of a prime $p$ in $n!$

Vedclass · Accepted Answer

$100$

Vedclass · Answer

(C) To find the largest power of $2$ that divides $\frac{200!}{100!}$,we use Legendre's Formula,which states that the exponent of a prime $p$ in $n!$ is given by $E_p(n!) = \sum_{k=1}^{\infty} \lfloor \frac{n}{p^k} \rfloor$.First,calculate the exponent of $2$ in $200!$:$E_2(200!) = \lfloor \frac{200}{2} \rfloor + \lfloor \frac{200}{4} \rfloor + \lfloor \frac{200}{8} \rfloor + \lfloor \frac{200}{16} \rfloor + \lfloor \frac{200}{32} \rfloor + \lfloor \frac{200}{64} \rfloor + \lfloor \frac{200}{128} \rfloor$$= 100 + 50 + 25 + 12 + 6 + 3 + 1 = 197$.Next,calculate the exponent of $2$ in $100!$:$E_2(100!) = \lfloor \frac{100}{2} \rfloor + \lfloor \frac{100}{4} \rfloor + \lfloor \frac{100}{8} \rfloor + \lfloor \frac{100}{16} \rfloor + \lfloor \frac{100}{32} \rfloor + \lfloor \frac{100}{64} \rfloor$$= 50 + 25 + 12 + 6 + 3 + 1 = 97$.The exponent of $2$ in $\frac{200!}{100!}$ is $E_2(200!) - E_2(100!) = 197 - 97 = 100$.Therefore,the largest power of $2$ is $100$.

The largest power of $2$ that divides $\frac{200!}{100!}$ is

Similar Questions

The number of $3$-digit numbers,formed using the digits $2, 3, 4, 5$ and $7$,when the repetition of digits is not allowed,and which are not divisible by $3$,is equal to ..........

If ${ }^{n} P_{r}={ }^{n} P_{r+1}$ and ${ }^{n} C_{r}={ }^{n} C_{r-1}$,then the value of $r$ is equal to:

If all the letters of the word $ACADEMICIAN$ are permuted in all possible ways,then the number of permutations in which no two $A$'s are together and all the consonants are together is:

How many words can be formed by taking two identical and two distinct letters from the letters of the word $COMBINATION$?

Out of $8$ students in a classroom,$4$ of them are chosen and they are arranged around a table. If the remaining $4$ are arranged in a row,then the total number of arrangements that can be made with those $8$ students is

Vedclass Test Series

Exam Paper Generator

Online Exam Module