lenagmat wrote:Please could you explain how does it implies mn = 2 * 3 * (5^2) = 150 from 2^3 * 3^3 * 5^2 * mn = k^4
By itself it doesn't imply that exactly. The idea is that 5400mn=k^4 implies that 5400mn can be represented as some integer raised to the power of 4. For that to be true,
all of the exponents in the number's prime factorization must be multiples of 4. For example, 2^12*3^8*5^4=(2^3*3^2*5)^4=360^4. This is only possible because the exponents in the prime factorization(12,8,4) are all multiples of 4.
In this problem, 5400mn=2^3*3^3*5^2mn, so we need to choose mn so that it makes the exponents on the right side all multiples of 4. If mn=2*3*5^2, then the expression becomes 2^4*3^4*5^4=(2*3*5)^4=30^4. However, if we are only interested in making it a 4th power, we also could choose mn=2*3^5*5^6, so the expression becomes 2^4*3^8*5^8=(2*3^2*5^2)^4=450^4. We have infinitely many choices for mn to make 5400mn equal to an integer raised to the fourth power. However, in this problem we also want to minimize m+n which implies that we should choose the lowest possible value of mn.
