Five consecutive positive integers have the property that the sum of the second, third and fourth is a perfect square, while the sum of all five is a perfect cube. If m is the third of these five integers, then the minimum possible value of m satisfies

(A) m ≤ 200
(B) 200 < m ≤ 400
(C) 400 < m ≤ 600
(D) 600 < m ≤ 800
(E) m > 800

• We know that m being the third term (Middle term) is the 'mean or average' of these 5 consecutive integers so we can say-
  Sum of 5 integers= 5*third term
  --> Sum= 5m which ATQ is a perfect cube
  
  Similarly, Sum=3m is a perfect square
  
  So minimum possible value of m that satisfies both the conditions will be Cube of 3 x Square of 5 = 27x25= 675. Therefore D is the right option.
• 8 years agoHelpfull: Yes(0) No(0)