Divide and Rule

Number Theory Level pending

Let $x_1,x_2,\cdots,x_n$ be integers such that

$(x_i-3)|(x_i^3-3) , \forall i=1,2,\cdots,n$

Find $\sum_{i=1}^n x_i$ .

1 solution

Janardhanan Sivaramakrishnan
Jan 19, 2015

Let $y=x-3$ . Then the problem is equivalent to

$((y+3)^3-3) \mod y = \left(y(y^2+9y+27)+27-3\right) \mod y = 0$

$24 \mod y = 0$

Or, $y$ divides 24. Possible choices are $y=\pm 24, \pm 12, \pm 8, \pm 6, \pm 4, \pm 3, \pm 2, \pm 1$ .

Consequently, $x=y+3=-21,27,-9,15,-5,11,-3,9,-1,7,0,6,1,5,2,4$ .

Thus, the required sum is $-21+27-9+15-5+11-3+9-1+7+0+6+1+5+2+4=\boxed{48}$