# Playing with Powers of 2

Number Theory Level pending

Let $$P\left( x \right)$$ be the sum of the digits of $$x$$. We define $$T\left( x\right)$$ as applying $$P\left( x \right)$$ to the number $$x$$, and then to the result (repeatedly), until the outcome is a single digit. For example, $$T\left( 249 \right) :\quad P(249)=15,\quad P(15)=6$$ so $$T\left( 249 \right) =6$$.

Let $$T({ 2 }^{ 2016 })=N$$. Find the least possible value for $$x>1$$ that satisfy $${ 2 }^{ P(x) }\equiv N \pmod x$$.

×