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

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(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

$100$

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(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

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