# Bringing A whole new level

**Number Theory**Level 5

For every positive integer \(g\) there is a unique factorial base expansion \((g_1,g_2,g_3,\ldots,g_m)\), meaning that

\(g=1!\cdot g_1+2!\cdot g_2+3!\cdot g_3+\cdots+m!\cdot g_m\), \(\text{where each}\) \(g_i\) is an integer, \(0\le g_i\le i\) , and \(0<g_m\).

Given that \((g_1,g_2,g_3,\ldots,g_j)\) is the factorial base expansion of \(16!-32!+48!-64!+\cdots+1968!-1984!+2000!\), find the value of \(g_1-g_2+g_3-g_4+\cdots+(-1)^{j+1}g_j\).