Actually the answer is 2, but that's not the answer your teacher
expects.  But it is correct.  If you divide 2 by any integer larger
than it is, you get the remainder 2.

  0     0     0     0     0
4)2   6)2   8)2  12)2  16)2
  0     0     0     0     0
  2     2     2     2     2   

So it always leaves a remainder of 2.

But I'll bet your teacher expects one larger than 2 itself.

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So Let the number be N

which is the least number > 2 that when divided by 4,6,8,12,&16 leaves a
remainder of 2

So 

N = 4a+2 = 6b+2 = 8b+2 = 12c+2 = 16d+2

Subtract 2 from each of those

N-2 = 4a = 6b = 8b = 12c = 16d

Therefore N-2 must be a multiple of 4,6,8,12, and 16, so it is
the least common multiple of 4,6,8,12, and 16.

  4 = 2×2
  6 = 2      ×3
  8 = 2×2×2
 12 = 2×2    ×3
 16 = 2×2×2×2 
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LCM = 2×2×2×2×3 = 48

The least common multiple of those is 48

So N-2 = 48

Therefore N = 50.

Edwin