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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?

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    [general solution](https://docs.google.com/open?id=0B9GIk5AfjFmSNWE1ZGE5M2UtOWYxYi00MjFkLWIyOTktYTRkZTY4ODNmNWZk)2012-02-10

2 Answers 2

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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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Some (less than beautiful) Maple code to do the brute force search: (putting X's in for unknown digits)

mystery_number := "303X1625XX96"; mystery_number := convert(mystery_number,list): digits := nops(convert(mystery_number,list)): k := ceil(digits/6): # since we have m is approx n^6  for n from 1 to 10^k do   flag := true;   m := (n*(n+1)*(n+2))^2;   nList := convert(convert(m,string),list);   if nops(nList) = digits then       for i from 1 to digits do         if flag and (mystery_number[i] = nList[i] or mystery_number[i] = "X") then             flag := true;          else             flag := false;          end if;      end do:      if flag then print(m); end if;   end if; end do: 

For example: Starting with "XXXXXXXXXX9X" the only 2 solutions are 303916253796 and 646840581696

In fact, merely knowing that the desired number is 12 digits long (using "XXXXXXXXXXXX") produces a list of 32 possibilities. So analyzing the number of digits might be a very good place to start.