The age numbers of four people are a sequence of natural numbers. The oldest person in these four people is not older than 30. The product of four ages number is divisible by 2700, but not 81. How old are these four people？
My idea： Since they are a sequence of natural numbers, Let these four numbers are: (x-2),(x-1), x ,(x+1) for $ x+2 \le 30 $
$$ The product = (x-2)(x-1)x(x+1) $$ $$ =x^4 - 2x^3 -x^2 +2x $$
since the product is divisible by 2700，but not 81. We know $$2700 = 2^2 * 5^2 * 3^3 , 81 = 3^4 $$ Thus, $$ x^4 - 2x^3 -x^2 +2x = 2^2 * 5^2 * 3^3 * z $$ for {z | z cannot be 0，3 or any numbers that comes from 3 times a natural number｝.
i think that the solution might apply "2700 is the least common multiple of the product and ... ",I am stucked here and have no idea for the next step.
Since you’re only taking $4$ consecutive numbers, only one can be a multiple of $5$. But $25$ divides the product, so your multiple of $5$ must be a multiple of $25$, and the only multiple of $25$ that’s less than $30$ is $25$.
You must have exactly $3$ factors of $3$ in your product. You can only have at most two multiples of $3$ among $4$ consecutive integers, so for $27$ to divide the product, you need at least one multiple of $9$. The only multiple of $9$ close enough to $25$ is $27$.
Since $81$ does not divide the product, $27$ needs to be the only multiple of $3$ among your integers. Thus, they must be $25, 26, 27, 28$.