If a is divisible by 2, is b + 5 an integer?

I bolded two parts of your explanation. You originally indicated that 2k was even. k itself may be even or odd. You later asserted that k was even.

I don't think the given OA is correct. Anyways, let a = 2p,where p is any integer.

From F.S 1, we have that\(\frac{(a+b)}{2}\) = Not an integer

or\(\frac{2p}{2}+\frac{b}{2} = p+\frac{b}{2}\)= Not an integer. Assuming, b = 3 , we satisfy the fact statement and get a YES for the question stem. Again assuming b=1.2, we satisfy the fact statement but get a NO for the question stem.Insufficient.

From F.S 2, we have \(3a+b+(b+10)/3\) = 2k, where k is again an integer.

Thus,\(3a/3 + (b+5)/3+(b+5)/3\) = 2k

or 2/3*(b+5) = 2k-a = 2k-2p = 2(k-p)[an integer]

or (b+5) = An integer*3 = integer.

Sufficient.

B.

From the stem we cannot assume that b is a rational number, it could be an irrational as well, for example \(\sqrt{2}\).

On the GMAT, if no other constraint is given about a variable, then all we can say that it's a real number.

If a is divisible by 2, is b + 5 an integer?

Notice that:
a is divisible by 2 implies that a is an even number.
The question basically asks whether b is an integer.

(1) The median of a and b is not an integer. b may or may not be an integer. Consider: a=2 and b=3 for an YES answer and a=2 and b=1/2 for a NO answer. Not sufficient.

(2) The average (arithmetic mean) of 3a, b, and b + 10 is an even integer --> \(3a+b+(b+10)=3*even\) --> \(2b+10=3*even-3a\) --> since a is even, then \(2b+10=3*even-3*even\) --> \(2(b+5)=3(even-even)=3*even\) --> \(b+5=3*\frac{even}{2}=3*integer=integer\). Sufficient.

Answer: B.

Hope it's clear.

You make a good point. In any case, it's not an integer.

a = even

b+5 is an integer?

(1) The median of a and b is not an integer.

a+b/2 is not an integer.

Its given that a is even. The above expression can not be an integer if b is odd or fraction.

(2) The average (arithmetic mean) of 3a, b, and b + 10 is an even integer

3a+b+b+10/2= integer

3a+2b+10/2= integer

3a is even (a is even), 2 b is even and 10 is even. To have above equation as an integer b must be an integer.

Hence b+5 will be an integer.

B is the answer

From the question stem it is clear that \(a\) is \(Even\) and we need to know whether \(b\) is integer

Statement 1: implies \(\frac{a+b}{2} =\) Non Integer (NI). if \(a=2\) & \(b=1.5\), then median is non integer and \(b+5\) is also a non integer

but if \(a=2\) & \(b=3\), then median is non integer but \(b+5\) is an integer. Hence we have both a Yes & a NO. Insufficient

Statement 2: implies \(\frac{3a+b+b+10}{3}=Even => 3a+2b+10=3*Even\)

\(=>2b=Even-10-3a\). Now as \(a\) is even so \(3a\) is even and \(10\) is also even

so \(2b=Even-Even-Even=Even => b=\frac{Even}{2}=Integer\). Hence \(b+5\) will be an integer. Sufficient

Option B

I have a major confusion here.

Everyone here is assuming that a is an int. A number divisible by 2 can be a non-integer as well. Ex - 2.2

Every GMAT divisibility question will tell you in advance that any unknowns represent positive integers (ALL GMAT divisibility questions are limited to positive integers only). So, you are right, a proper GMAT question should have explicitly mentioned that a is a positive integer. But as far as GMAT is concerned, a is divisible by b means that:

(i) a is an integer;
(ii) b is an integer;
(iii) \(\frac{a}{b}=integer\).

So, 2.2 is divisible by 2 does not make sense because 2.2 is NOT an integer.

Hope it helps.