guerrero wrote:If a and b are positive integers, and (2^3)*(3^4)*(5^7) = (a^3)*b, how many different possible values of b are there?
Note that number of possible values of b will be same as the number of possible values of a^3.
The left-hand side has three 2s, four 3s, and seven 5s.
As the right-hand side contains a^3, a^3 must contain 2 and/or 3 and/or 5 in multiples of 3.
In other words a^3 can be equal to 1 or (2^3) or (2^3)*(3^3) etc.
Hence, number of possible values of a^3 = (Number of ways to select 2 in multiples of 3)*(Number of ways to select 3 in multiples of 3)*(Number of ways to select 5 in multiples of 3)
Number of ways to select 2 in multiples of 3 : In 2 ways 2^0 or 2^3
Number of ways to select 3 in multiples of 3 : In 2 ways 3^0 or 3^3
Number of ways to select 5 in multiples of 3 : In 3 ways 5^0 or 5^3 or 5^6
Hence, number of possible values of a^3 = 2*2*3 = 12
So, number of different possible values of b = 12
The correct answer is E.
