Write down all three – digit numbers made of digits 0, 2, and 5 only (may be not all at the same time) , that are: Multiples of 2

Question

 No answer that I have figured out yet.

Answer 1

To list the numbers that fit the constraint, “all three – digit numbers made of digits 0, 2, and 5 only,” consider a counter (with only three values, not 10, per position):
First Second Third 
  0       0        0
  2       2        2
  5       5        5

There are only 27 (that is, 3*3*3) of these.

Since numbers (considered as decimal) with the ones digit equal to either 0 or 2 are even, that means that the “counter” only “clicks” using these values:
First Second Third
  0      0         0
  2      2         2
  5      5

That makes 18 (that is, 3*3*2) [possibilities. They may enumerated easily just by incrementing the counter:
First Second Third
  0      0         0
  0      0         2
  0      2         0
  0      2         2
  0      5         0
  0      5         2       *
  2      0         0
  2      0         2
  2      2         0
  2      2         2
  2      5         0       *
  2      5         2
  5      0         0
  5      0         2       *
  5      2         0       *
  5      2         2
  5      5         0
  5      5         2

 

Now, you will must eliminate the numbers that have all three digits at the same time ["may be not all at the same time)].  These have a "*" in the list above.

 

To visualize how these numbers are selected, think of picking either 0 or 2 as the ones digit, then having two remaining choices for the tens digit, then one remaining choice for the hundreds digit.  That makes 4 (that is, 2*2*1 ) numbers.  Showing a special counter (it skips digits already used):

First Second Third

  5      2         0

  2      5         0

  5      0         2

  0      5         2

 

Remove these four values from the list of 18 and you are left with 14.