SOLUTION: the integer 33 is to be expressed as a sum of n consecutive positive integers. The value of n could be which of the following ? I. 2 II. 3 III. 6 (A) I only (B)

The sum of consecutive integers is an arithmetic series
with d=1 and first term 
 

 
Substituting d=1 and =33
 

 
Multiplying through by 2
 

 

 

 

 
There are only 4 possible factorizations possible for that
so that the middle term's coefficient will be positive: 
 
1.  (n+66)(n-1)=0 which has middle term 65n
 
So n=1.  We can't have just one "consecutive" integer, for
there has to be at least two for each to be "consecutive" to.
 
2.  (n+33)(n-2)=0 which has middle term 31n

So 
   
   
 
and n=2, and the 2 consecutive integers are 16+17=33

3.  (n+22)(n-3)=0 which has middle term 20n

So 
   
   

and n=3, and the 3 consecutive integers are 10+11+12=33


4.  (n+11)(n-6)=0 which has middle term 5n

So 
   
   

and n=6, and the 6 consecutive integers are 3+4+5+6+7+8=33

So there are three possibilities for n, 2, 3, and 6.

That's all three I, II and III which is choice (E)

Edwin