In a class, the teacher wrote a set of consecutive integers beginning with 1 on the blackboard. Little Johnny came and erased one number. The average of the remaining numbers was 2741327413. What was the number that Little Johnny erased? Choose the correct choice from the following answers:
A) 7 B) 9 C) 11 D) 13 E) 15
We have S = the sum of all integer from 1 to N   x = the missing integer   So as sum = average * Nb of terms   S-x= 27(4/13) * (N-1) (A)   As S-x should be an integer N-1 should be a multiple of 13.   So we have some possible scenario   N-1= 13 =&gt; N=14   N-1= 26 =&gt; N=17   N-1= 39 =&gt; N=40   N-1= 52 =&gt; N=53   ....   Back to our formula (A)   x = S - 27(4/13)*(N-1)   x = (N(N+1))/2 - 27(4/13)*(N-1)   As x&gt;0   (N(N+1))/2 &gt; 27(4/13)*(N-1)   approx 27(4/13) by 27   (N(N+1))/2 &gt; 27*(N-1)   N(N+1) &gt; 54*(N-1)   So we see than N=53 is working (indeed 53*54&gt;54*52)   So if N=53:   x = (53(54))/2 - 27(4/13)*(52)   x = (53(54))/2 - (27+4/13)*(52)   x = (53(54))/2 - 27*52-(4/13)*52   x = 53*27 - 27*52-(4/13)*52   x = 53*27 - 27*52-4*4   x = 27*(53-52) - 16   x = 27 - 16= 
<gadget id="calculator">27 - 16</gadget>
<output>11</output>
11   ANSWER:11
<result>C</result>
In a class, the teacher wrote a set of consecutive integers beginning with 1 on the blackboard. Little Johnny came and erased one number. The average of the remaining numbers was 2741327413. What was the number that Little Johnny erased?