A professor writes N consecutive natural numbers, beginning with 1, on the blackboard. One of the students in the class deletes one of the numbers (exactly one number), from that list.
Now, given that the average of the remaining N-1 numbers is 271/16.
Can you find out the number that has been deleted from the list ?
(In reply to
Answer by K Sengupta)
Let the deleted number be m(say)
Now, the sum of the first N consecutive natural numbers is N(N+1)/2
By the problem, the sum of the remaining N terms with the exclusion of m is (271/16)*(N-1)
Thus, (271/16)*(N-1) = N(N+1)/2 - m .......(i)
Since the lhs of (i) is an integer, it follows that N-1 is divisible by 16, so that: N=16t+1, for nonnegative integer values of t.
Then, m = (16t+1)(8t+1) -271t
= t(128t-247) +1 .......(ii)
And, N-m = (16t+1)-{t(128t-247)+1}
= t(263-128t) .........(iii)
Now, N> m
Since t is nonnegative, therefore, from (iii), we must have:
263-128t>=0
=> 128t<= 263 <384
=> t<3, so that:
However, substituting t=1 in (ii), we obtain:
m= (1)*(-119)+1 =-118<0, which is untenable.
Similarly, if we substitute t=0, then from (i), we have: m=1, and: N=1*2/2=1, so that: N=m, which is untenable.
Substituting t=2, we have:
N= 16*2+1=33
and, m= 2*9+1=19
Consequently, the required deleted number is 19.
Edited on January 27, 2022, 1:45 am
Puzzle Solution: Method I
