Let S = the sum of the 4 consecutive odd integers and P = the required perfect square.The sum of 4 consecutive two-digit odd integers, when divided by 10, becomes a perfect square. Which of the following can possibly be one of these 4 integers?
A. 21
B. 25
C. 41
D. 67
E. 73
The sum of 4 consecutive two-digit odd integers, when divided by 10, becomes a perfect square.
S/10 = P
S = 10P.
In other words, the SUM of the 4 consecutive odd integers is 10 times a perfect square.
Options for P:
1, 4, 9, 16, 25, 36...
Options for 10P:
10, 40 , 90, 160, 250, 360...
The value of S is contained in the list above.
If x is the smallest of the 4 consecutive odd integers, then S = x + (x+2) + (x+4) + (x+6) = 4x + 12 = 4(x+3).
Implication:
The value of S must be a multiple of 4.
Of the values in the list above, only the following are multiples of 4:
40, 160, 360.
If S = 40, then the average of the 4 integers = 40/4 = 10.
If S = 160, then the average of the 4 integers = 160/4 = 40.
If S = 360, then the average of the 4 integers = 360/4 = 90.
The numbers in the answer choices are too great to yield an average of 10 and too small to yield an average of 90.
Thus, the required average is 40, implying that S is equal to the following:
37+39+41+43 = 160.
The correct answer is C.
