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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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1  
Find prime factors of $\sqrt k$. –  Lazar Ljubenović Feb 10 '12 at 8:41
    
Your result is 11 digit, so running a program from approximation of $k^{1/3}$ is not bad. –  Saeed Feb 10 '12 at 8:47
4  
The equation [n(n+1)(n+2)]^2=k isn't exactly what we have here because $k$ in the stated problem is almost but not fully known. But for computational purposes, the facts (a) $n\approx k^{1/6}-1$, (b) $k$ is a square, (c) $36|k$, and (d) $n|(k-4)$ should be all we need for not-too-large numbers. –  anon Feb 10 '12 at 8:47
    
general solution –  pedja Feb 10 '12 at 8:55

2 Answers 2

up vote 5 down vote accepted

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.

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