2021 GMC 10B Problems/Problem 18

Problem

Let $f(n)$ be the largest possible power of $2$ that divides $n$ . Find $f((3^2-3)(4^2-4)(5^2-5)(6^2-6)(7^2-7)(8^2-8)...(99^2-99)(100^2-100))$ .

$\textbf{(A)} ~191 \qquad\textbf{(B)} ~192 \qquad\textbf{(C)} ~193 \qquad\textbf{(D)} ~198\qquad\textbf{(E)} ~199$

Solution

Note that $f(n) = \nu_2 (n)$ , where $\nu_2(k)$ is the $2$ -adic valuation of $k$ . By LTE, $\begin{align*} f((3^2-3)\dotsm(100^2-100)) &= f(3^2 - 3)+\dotsb + f(100^2 - 100) \\ &= f(2(3))+ \dotsb +f(99(100)) \\ &= f(2) + f(3) + f(3) + \dotsb + f(99) + f(99) + f(100) \\ &= f(2) + f(100) + 2\sum_{n = 3}^{99}f(n). \end{align*}$

To evaluate the sum, we use casework on the divisibility of $n$ over $2^n.$ For example, for $n=1$ , we count the numbers from $3$ to $99$ which are divisible by $2$ . $n = 1$ : $48$ numbers, $n = 2$ : $24$ numbers, $n = 3$ : $12$ numbers, $n = 4$ : $6$ numbers, $n = 5$ : $3$ numbers, $n=6$ : $1$ number, so adding, we get $\sum_{n=3}^{99} f(n) = 94$ , and finishing, $\begin{align*} f((3^2-3)\dotsm(100^2-100)) &= 3 + 2(94) \\ &=\boxed{\textbf{(A)}~191}\hskip 0.5pt. \end{align*}$ ~pineconee

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2021 GMC 10B Problems/Problem 18

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