A positive integer n is called a “good” number if n3 + 7n − 133 = m3 for
some positive integer m. What is the sum of all “good” numbers? Select:
A) 24 B) 25 C) 29 D) 30 E) 0
We consider the size of m relative to n. If m = n, then n is a good number if and only if   7n = 133. In this case, we get 19 is a good number. Now, suppose that m n + 1. Then   n3 + 7n − 133 = m3 (n + 1)3 = n3 + 3n2 + 3n + 1   so that 3n2 − 4n + 134 0. For positive integers n, it is easy to see that this is impossible. It   remains to consider m n − 1. For such m, we have   n3 + 7n − 133 = m3 (n − 1)3 = n3 − 3n2 + 3n − 1   which implies 3n2 + 4n − 132 0. It follows here that n 6. One checks that   63 + 7 · 6 − 133 = 53 and 53 + 7 · 5 − 133 = 33,   so 6 and 5 are good numbers. For n 4, we see that n3 +7n−133 &lt; 0 and, hence, n3 +7n−133   cannot equalm3 for a positive integerm. Therefore, the sum of all good numbers is   19+6+5= 
<gadget id="calculator">  19+6+5</gadget>
<output>30</output>
30.   correct answer 30
<result>D</result>
A positive integer n is called a “good” number if n3 + 7n − 133 = m3 for
some positive integer m. What is the sum of all “good” numbers?