Prove the divisibility of the following numbers:

question a

45^45·15^15 by 75^30

question b

24^54·54^24·2^10 by 72^63

question c

45^10·5^40 by 25^20

question d

12^8·9^12 by 6^16

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)

(B)

In this case, it is easier to also factor the divisor to primes:

Both of these factor must be matched in the dividend in order to prove divisibility, and that indeed turns out to be true: