lean_workbook_plus_59538

We have $v_2(3^{101}+5^{101})=v_2(3+5)+v_2(101)=3+0=\boxed{3}.$ If you don't know LTE: \nWe have $3^{101}+5^{101}=(3+5)(3^{100}-3^{99}5+\cdots+3^{50}5^{50}-\cdots-5^{99}3+5^{100}).$ The second term has an odd number of odd terms( $101$ ), so it is odd. The only factors of $2$ are in $3+5=8\implies\boxed{3}.$