1962 AHSME Problems/Problem 36

Problem

If both $x$ and $y$ are both integers, how many pairs of solutions are there of the equation $(x-8)(x-10) = 2^y$ ?

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \text{more than 3}$

Solution

It is obvious that x>10 , thus x=12 , y=3. Then, let $x-8=2n$ , $x-10=2n-2$ , it can be written as $2n*(2n-2) = 2^y$ , Also, $n*(n-1) = 2^y-2$ , so, n only can be 2 , y=3, and the answer is $\boxed{B}$ .

Page

Toolbox

Search

1962 AHSME Problems/Problem 36

Problem

Solution