Question 1165100
"Integers" includes negative values, unless otherwise restricted.

-1 + 0 + 1 = 0  
0 + 1 + 2 = 3    (3 is the smallest positive integer that is the sum of three consecutive integers)<br>

-1 + 0 + 1 + 2 = 2  (2 is the smallest positive integer that is the sum of four consecutive integers)<br>

etc.

5 consec. integers:  min sum gt zero = 5<br>
7 consec. integers:  min sum gt zero = 7<br> 

<br>
(b) if  N = a + (a+1) + (a+2) + (a+3) = 4a + 6
and   N = b + (b+1) + (b+2) + ... + (b+5) = 6b + 15<br>

then     6b + 15 = 4a + 6
          6b + 9 = 4a
         and  a = (6b + 9) / 4    

which is not an integer (6b is even, even + odd is odd, odd divided by 4 has a remainder) <br> 

   
To the student:  Oh, I may have misread the question.  I see what you mean... perhaps the question should have been "Find the smallest positive integer which can be written as the sum of three, four, five, AND seven consecutive numbers."   but OR makes sense too upon re-reading.