M22-22

Official Solution:


Is \(a+b+c\) an even integer?

It's important to note that we're not provided with information that \(a\), \(b\), and \(c\) are integers.

(1) \(a-c-b\) is even.

If all the variables are integers, then \(a+b+c\) will indeed be even. However, if they're not, for instance if \(a=3.5\), \(b=1\), and \(c=0.5\) (such that \(a-c-b=2\), which is even), then \(a+b+c=5\), which is odd. Not sufficient.

(2) \(\frac{a-c}{b}\) is odd.

The scenario is similar to the first one: if the variables are integers, then \(a+b+c\) will be even, but if they're not, for example if \(a=3.5\), \(b=1\), and \(c=0.5\) (in this case \(\frac{a-c}{b}=3\), which is odd), then \(a+b+c=5\), an odd number. Not sufficient.

(1)+(2) The sum \(a+b+c\) could be even or odd (again, if the variables are integers, the answer would be YES, but if \(a=3.5\), \(b=1\), and \(c=0.5\), the answer is NO). Thus, the combination of conditions is still not sufficient.


Answer: E