Another Consecutive number question

[milanproda]

Three consecutive numbers are drawn from integers, which are strictly greater than 9 and strictly less than 20. Suppose w is the product of the numbers drawn, which of the following must be true?

I-w is an integer multiple of 3
II- W is an integer multiple of 4
III-w is an integer multiple of 6

Answers:
A-I ONLY
B- II ONLY
C- I & III ONLY (ANSWER)
D-II & III ONLY
E- I, II, & III

I tried picking this question apart without doing any math. I saw that the w must be even (because one of the three consecutive numbers was even), but I could not discern anything past that.

Eventually I plugged in numbers, starting from 10x11x12, 11x12x13...The product of the first three sets of numbers were al divisible by 3, 4, and 6. Only later sets were not divsible by 4. I was able to get the answer but after taking about 4 minutes to figure it out. Is there a simpler way to answer questions like this and figure out the variable?

[Tani]

Hi Milan,

You are correct that one must be even, but one also must be a multiple of three since every third number is a multiple of three. Therefore I and III are correct. However, the product will only be a multiple of four if the group either includes a multiple of four (12 or 16), or starts and ends with an even number. So two groups, 13, 14, 15 and 17, 18 19 will be divisible by 2 and three, but not by four.

[milanproda]

Tani, thank you again!

I did not know that the product of every third consecutive number must be a multiple of 3. Are there any other rules similar to that one?

Also, just becuase W is definately a multiple of three, how could I be sure that it is also a multiple of 6 (for example, 33 is a multiple of 3 yet 6 is not). If you can, please let me know where I am wrong.

[Tani]

It works for every integer - starting with 2 - every other number is divisible by two
starting with three - every third number is divisible by three
starting with 4.... etc.

starting with 17, every seventeenth number is divisible by 17 and on and on...

That's because to get multiples of a number you simply keep adding that number to the prior multiple