The problem says "The product of integers x, y and z is even. Is z even?"

1) x/y = z
2) z= xy

In the back of the book it says that from statement 1, x/y=z we know that x and y are even. How so?

Also, how is statement 1 not sufficient if statement 2 is? Aren't they basically the same principle? for instance, z = xy and xyz= even, is the same as saying (xy)(xy) = even, yes? So how is x = yz, from statement 1, not the same as (yz)(yz) = even?

Thanks,
Sarah

Sarah,

x * y * z = even there are 3 ways for this to happen:

- all three numbers are even
- two of the three numbers are even
- one of the three numbers is even

the only way for x * y * z to not be even is if all three are odd

from (1), x = yz ... x can't be odd, because that would force y and z to be odd and, per
the above, all three numbers can't be odd

however x could be even if y and z are both even or if one or the other is odd, so z
could be even or odd. So (1) is not sufficient. x could be even if y is odd and z is even, so x and y need not both be even.


(2) is sufficient since z = xy, z can't be odd since z being odd would force x an y to
both be odd and, per the above, all three can't be odd

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