Former Math Kangaroo problem

Number Theory Level 2

Ann has written 2012 as $m^{m}(m^{k} - k)$ for some positive integers $m$ and $k$ . What is the value of $k$ ?

1 solution

Chris Sapiano
Jun 8, 2019

Prime factorisation of 2012 is $2^2 * 503$ and therefore $m = 2$ .

Substitution $m$ as $2$ , we get: $2^k - k = 503$

$2^k = 503 + k$

$2^9 = 512$ and therefore $k = \boxed{9}$