<s>[INST] Given that $[n(n+1)(n+2)]^2 = 303916253\square96$, find the value of $\square$. 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?
 [/INST] Then, there's brute force:
303,916,253,?96 ~= 30.3 e+10 ~= (5.5 e+5) squared ~= (n cubed) squared.
So, 80 < n < 90 since 80 cubed = 512,000 and 90 cubed = 729,000.
Since 303,916,253,?96 ends with 6, the number that is squared must end with 4 or 6.  Hence, none of n, n+1, and n+2 can have 0 or 5 in the one's digit.
This limits possible choices to 81*82*83 or 82*83*84 or 86*87*88 or 87*88*89. 
As it turns out, with n=81, 81*82*83 = 551,286, which when squared = 303,916,253,796.
The missing digit (?) is 7.
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