Given that $[n(n+1)(n+2)]^2 = 303916253\square96$, find the value of $\square$.
Given that $[n(n+1)(n+2)]^2 = 30391625\square796$, find the value of $\square$.
Problem
Given that $[n(n+1)(n+2)]^2 = 3039162537\square6$, find the value of $\square$.
Solution
In any three consecutive integers, $n, n+1, n+2$, at least one of the numbers will be even, and one of them will be a multiple of 3. Hence the product, $n(n+1)(n+2)$, will be even and divisible by 3. Furthermore, the square of an even number will be divisible by 4, and the square of a multiple of 3 will be divisible by 9. If a number is divisible by 9, the sum of the digits will also be divisible by $9: 3 + 3 + 9+ 1 + 6 + 2 + 5 + 3 + 7 + 6 = 45$, so the value of $\square$ must be 0 or 9. However, if the number is divisible by 4, the last two digits (either 06 or 96) must be divisible by 4. Hence the value of $\square$ is 9.
Find the value of n. How would you solve the equation, $[n(n+1)(n+2)] ^2 = k$, in general?