It can't be divided by 5 or 10 since when you count by 10's the numbers end in only zeros and when you count by 5 it is either 5 or 0 (5, 10, 15, 20 and so on).
9 will only work if the digits in the answer add up to 9. 18: 1+8=9
99: 9+9=18 and 1+8=9 So it can't be divided by 9 102: 1+0+2=3 (not 9)
That leaves 6 and six works - here's why.
There is a rule for 3's that says that the sum of the digit must add up to 3, 6 or 9.
Look:
3x1 = 3
3 x 2 = 6
3 x 3 = 9
3 x 4 = 12 and 1+2=3
3 x 5 = 15 and 1+ 5 = 6
3 x 6 = 18 and 1+ 8 = 9
3 x 7 = 21 and 2 +1 = 3 and so on - do you see the 3,6,9 pattern repeating.
Six follows the same rule except the number must be even. Look above to see that 6 works with 6, 12 and 18 but not the others.
So 102 is divisible by 3 (1+0+2 = 3) AND it is an even number - therefore it is divisible by 6. Here's proof: 6 x 15 = 90 and 102 is 12 more than 90 and six goes into that 12)
I hope that this will help you feel better. I don't want you to cry. :-)