Question

The integer 90can be expressed as the sum of z consecutive integers. The value of z could be any of the following except

A) 3
B) 4
C) 5
D) 6
E) 9

Answer 1

If the first integer is n+1, the consecutive integers are: n+2, ...n+z.

If 90 is the sum of these z consecutive integers then...

90 = sum{i=1 to z} n+i
90 = zn + sum{i=1 to z} i
90 = zn + z(z+1)/2

.: n = (180-z(z+1))/2z

Testing the given values of z:
A) z=3 => n = (180-3*4)/6
B) z=4 => n = (180-4*5)/8
C) z=5 => n = (180-5*6)/10
D) z=6 => n = (180-6*7)/12
E) z=9 => n = (180-9*10)/18

One of these 'n' is not an integer.
One of these 'n' does not belong.
Can you tell me which one?

Answer 2

(D)6

6x + 5*6/2 = 90
6x + 15 = 90
6x = 75
x is not an integer