How to find $\gcd(100! , 2^{100})$

Question

How can I find $\gcd(100! , 2^{100})$? If I use this method: $$100! = 2^x + y$$ How can I find $x$?

Answer 1

You have, applying Legendre's formula, and denoting $v_2(n)$ the exponent of $2$ in the prime factors decomposition of $n$: $$v_2(100!)=\biggl\lfloor\frac{100}2\biggr\rfloor+\biggl\lfloor\frac{100}4\biggr\rfloor+\biggl\lfloor\frac{100}8\biggr\rfloor+\dotsm$$