Hi

four consecutive integers 

Letting x,(x+1)(x+2)(x+3) represent them.  Their sum is 4x+6

Question states***

 51 <4x + 6 <65   OR    11 < x < 15   

12,13,14,15  OR 14,15,16,17 are four consecutive integers whose sum is between 51 and 65

CHECKING our Answer***

  12+13+14+15 = 54

  14+15+16+17  = 62 

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