Question 1150758
Let {{{n}}} be the {{{first}}} number, then the {{{last}}} is {{{n+566}}}, and the average is {{{(n + n + 566)/2 = n + 283}}}.

The sum is {{{567(n + 283) = (7 * 3^4)(n + 283)}}}

Since the {{{cube}}} has to have {{{7}}} and {{{3}}} as a factor, the smallest value will be when

{{{n + 283 = 7^2* 3^2 = 441}}}

That "tops up" the factors of {{{7}}} and {{{3 }}}to a multiple of {{{3}}}, so that you get a {{{perfect}}}{{{ cube}}}.

The cube is {{{567 * 441 = 7^3 * 3^6 = 63^3= 250047 }}}.