N is a positive integer. Is N even?
1) Sum of numbers from 1 to N is even.
2) Sum of numbers from 1 to 2N is even
1) Sum of numbers from 1 to N is even.
2) Sum of numbers from 1 to 2N is even
N is a positive integer. Is N even?
1) Sum of numbers from 1 to N is even.
2) Sum of numbers from 1 to 2N is even
Condition: N is a +ve Integer.
Statement-1:
Sum of the numbers from 1 to N == (N*(N+1))/2.
Condition given is sum of those numbers should be EVEN. We can re-write as,
(N*(N+1)/2) = EVEN.
If N=EVEN, then N/2 would be EVEN, then N+1=EVEN+1=ODD. Then N/2 * (N+1) = EVEN * ODD == EVEN.
If N=ODD, then (N+1)= ODD+1=EVEN and (N+1)/2= EVEN/2= EVEN, then N * (N+1)/2 = EVEN * ODD == EVEN.
So, however the N may be EVEN or ODD, then Statement-1 is NOT SUFFICIENT.
Statement-2:
Sum of the numbers from 1 to 2N == (2N*(2N+1))/2 = N * (2N+1).
Condition given is sum of numbers should be EVEN. We can re-write as,
If N=EVEN, then 2N+1=2(EVEN)+1=EVEN+1=ODD,So, N * (2N+1) = EVEN * ODD = EVEN
If N=ODD, then 2N+1=2(ODD)+1=EVEN+1=ODD, So, N * (2N+1) = ODD * ODD = ODD. Its not satisfying the condition.
From this, we could say that N should be +ve EVEN integer to satisfy the given condition in statement-2.
Hence, Statement-2 alone is SUFFICIENT.
Thus the correct answer is B.
HTH, GOOD LUCK,
Thanks,
Rajesh,
Loves GMAT....!!!!
