Power Of 2

${ 2 }^{ \alpha }+{ 2 }^{ \beta }+{ 2 }^{ \gamma }+.\quad .\quad .\quad .\quad .\quad .+{ 2 }^{ n }=10070000.\\ { 2 }^{ \alpha -4 }+{ 2 }^{ \beta -4 }+{ 2 }^{ \gamma -4 }+.\quad .\quad .\quad .\quad .+{ 2 }^{ n-4 }=629375.\quad \Rightarrow \quad \boxed { \alpha =4 } .\\ { 2 }^{ \beta -5 }+{ 2 }^{ \gamma -5 }+{ 2 }^{ \delta -5 }+.\quad .\quad .\quad .\quad .+{ 2 }^{ n-5 }=314687.\quad \Rightarrow \quad \boxed { \beta =5 } .\\ { 2 }^{ \gamma -6 }+{ 2 }^{ \delta -6 }+{ 2 }^{ \epsilon -6 }+.\quad .\quad .\quad .\quad .+{ 2 }^{ n-6 }=157343.\quad \Rightarrow \quad \boxed { \gamma =6 } .\\ { 2 }^{ \delta -7 }+{ 2 }^{ \epsilon -7 }+{ 2 }^{ \zeta -7 }+.\quad .\quad .\quad .\quad .+{ 2 }^{ n-7 }=78671.\quad \Rightarrow \quad \boxed { \delta =7 } .\\ { 2 }^{ \epsilon -8 }+{ 2 }^{ \zeta -8 }+{ 2 }^{ \eta -8 }+.\quad .\quad .\quad .\quad .+{ 2 }^{ n-8 }=39335.\quad \Rightarrow \quad \boxed { \epsilon =8 } .\\ { 2 }^{ \zeta -9 }+{ 2 }^{ \eta -9 }+{ 2 }^{ \theta -9 }+.\quad .\quad .\quad .\quad .+{ 2 }^{ n-9 }=19667.\quad \Rightarrow \quad \boxed { \zeta =9 } .\\ { 2 }^{ \eta -10 }+{ 2 }^{ \theta -10 }+{ 2 }^{ \iota -10 }+.\quad .\quad .\quad .\quad .+{ 2 }^{ n-10 }=9833.\quad \Rightarrow \quad \boxed { \eta =10 } .\\ { 2 }^{ \theta -13 }+{ 2 }^{ \iota -13 }+{ 2 }^{ \kappa -13 }+.\quad .\quad .\quad .\quad .+{ 2 }^{ n-13 }=1229.\quad \Rightarrow \quad \boxed { \theta =13 } .\\ { 2 }^{ \iota -15 }+{ 2 }^{ \kappa -15 }+{ 2 }^{ \lambda -15 }+.\quad .\quad .\quad .\quad .+{ 2 }^{ n-15 }=307.\quad \Rightarrow \quad \boxed { \iota =15 } .\\ { 2 }^{ \kappa -16 }+{ 2 }^{ \lambda -16 }+{ 2 }^{ \mu -16 }+.\quad .\quad .\quad .\quad .+{ 2 }^{ n-16 }=153.\quad \Rightarrow \quad \boxed { \kappa =16 } .\\ { 2 }^{ \lambda -19 }+{ 2 }^{ \mu -19 }+{ 2 }^{ \nu -19 }+.\quad .\quad .\quad .\quad .+{ 2 }^{ n-19 }=19.\quad \Rightarrow \quad \boxed { \lambda =19 } .\\ { 2 }^{ \mu -20 }+{ 2 }^{ \nu -20 }+{ 2 }^{ \xi -20 }+.\quad .\quad .\quad .\quad .+{ 2 }^{ n-20 }=9.\quad \Rightarrow \quad \boxed { \mu =20 } .\\ { 2 }^{ \nu -23 }+{ 2 }^{ \xi -23 }+{ 2 }^{ \o -23 }+.\quad .\quad .\quad .\quad .+{ 2 }^{ n-23 }=0\quad \Rightarrow \quad \boxed { \nu =23 } .\\ \\ So,\quad 2^{ 4 }+{ 2 }^{ 5 }+{ 2 }^{ 6 }+{ 2 }^{ 7 }+{ 2 }^{ 8 }+{ 2 }^{ 9 }+{ 2 }^{ 10 }+{ 2 }^{ 13 }+{ 2 }^{ 15 }+{ 2 }^{ 16 }+{ 2 }^{ 19 }+{ 2 }^{ 20 }+{ 2 }^{ 23 }=10070000.\\ \\ \therefore \quad (4+5+6+7+8+9+10+13+15+16+19+20+23)(13)=\boxed { 2015 } .(ANS)\\ \\$