If 2 is a factor of xyz, is...

[Guest]

If 12 is a factor of xyz, is 12 a factor of xy?

Answer: Cannot Be Determined.

I used the prime box to find the prime factors of 12: 2,2, and 3. According to the factor foundation rule, the products of these prime factors can divide into xyz. I get stuck right after this point. Aside from plugging in random numbers for xyz, how can I solve this problem? What's the easiest method?

Thank you for your time.

[rfernandez]

If 12 is a factor of xyz, is 12 a factor of xy?

You can definitely go about it using numerical evidence.

If xyz = 120 such that xy = 24 and z = 5, then yes, 12 is a factor of xy.
If xyz = 120 such that xy = 3 and z = 40, then no, 12 is not a factor of xy.

Another approach:
Knowing that xyz is divisible by 12, we can draw a prime box beneath xyz with 2, 2, 3, and a ?. (That ? is important -- it's a reminder that there may be other prime factors in xyz). Now, beneath xyz's prime box, draw three prime boxes, one for x, another for y and a third for z. Can you say anything definitively about which prime factors "live" in each of those three boxes? No. Therefore, we cannot make any conclusions about whether x and y have 2, 2, 3 in their combined prime boxes.