Question

Prove divisibility of these two numbers
`24^(54) * 54^(24) * 2^(10)` by
`72^(63)`

Answer 1

`72^(63)\\\\72=2\cdot2\cdot2\cdot2\cdot3\cdot3=2^4\cdot3^2\\\\72^(63)=(2^4\cdot3^2)^(63)=(2^4)^(63)(3^2)^(63)=2^(152)\cdot3^(126)`

`24^(54)\cdot54^(24)\cdot2^(10)\\\\24=2\cdot2\cdot2\cdot3=2^3\cdot3\\54=2\cdot3\cdot3\cdot3=2\cdot3^3\\\\24^(54)=(2^3\cdot3)^(54)=(2^3)^(54)\cdot3^(54)=2^(162)\cdot3^(54)\\\\54^(24)=(2\cdot3^3)^(24)=2^(24)\cdot(3^3)^(24)=2^(24)\cdot3^(72)\\\\24^(54)\cdot54^(24)\cdot2^(10)=2^(162)\cdot3^(54)\cdot2^(24)\cdot3^(72)=2^(162+24)\cdot3^(54+72)=2^(186)\cdot3^(126)\\\\=2^(34+152)\cdot3^(126)=2^(34)\cdot\underbrace{2^(152)\cdot3^(126)}_(72^(63))=2^(34)\cdot72^(63)`

`24^(54)\cdot54^(24)\cdot2^(10)=2^(34)\cdot72^(63)\\\\\text{therefore it is divisible by}\ 72^(63)`

`Used:\\\\(ab)^n=a^nb^m\\\\(a^n)^m=a^(nm)\\\\a^n\cdot a^m=a^(n+m)`