Answer:
Step-by-step explanation:
To prove divisibility, we need to factor the divident such that one of its factors matches the divisor.
(I use the notation x|y to denote that x divides y)
(A)
[tex]75^{30}|45^{45}\cdot15^{15}\\45^{45}\cdot15^{15}=3^{45}\cdot 15^{45}\cdot 15^{15}=\\=3^{45}\cdot 15^{60}=3^{45}\cdot 15^{30}\cdot 15^{30}=3^{45}\cdot (3\cdot5)^{30}\cdot 15^{30}=\\=3^{45}\cdot 3^{30}\cdot(5\cdot 15)^{30}=3^{45}\cdot 3^{30}\cdot(75)^{30}\\\implies\\75^{30}|3^{45}\cdot 3^{30}\cdot75^{30}[/tex]
(B)
[tex]72^{63}|24^{54}\cdot 54^{24}\cdot2^{10}\\24^{54}\cdot 54^{24}\cdot2^{10}=(2^{162}\cdot 3^{54})\cdot(2^{24}\cdot 3^{72}) \cdot 2^{10}\\=2^{196}\cdot 3^{126}[/tex]
In this case, it is easier to also factor the divisor to primes:
[tex]72^{63}=2^{189}\cdot 3^{126}[/tex]
Both of these factor must be matched in the dividend in order to prove divisibility, and that indeed turns out to be true:
[tex]2^{189}\cdot 3^{126}|2^{196}\cdot 3^{126}\implies\\2^{189}|2^{196}\,\,\mbox{and}\,\,3^{126}|3^{126}[/tex]
